ERRATA Page Line Should Read (iii) 9 u=(u,..,u ) 1 10 8 1. f. b. existence 10 6 f(t,x, u) 20 8 a + (f, ),.. 20 9 6> 0 23 13 f.b. (ix) 8 f.b. (x) 2 fo b. f +f Do.O 24 7 (xi) 7 f. b. Statement (x) applies 25 11 Fillipov 35 2 ly(t)-y(t')I < /2,.. 39 3 f.b. (t, x t1), t2, x(t2) ) 42 2 f. b....when A is not compact... 43 8 f. b. i is finite 2 f. bo tlk < t < t2k 2k 45 8 f. b. x(t) f (T,XkTU(T)UT)dT, t< t < t 45 8 f~b. xk (t): ftl o - 2k 46 7 K k kktitk 46 7. i(Yk Yk) +Yk = k + Yk 55 13.. and (ix) of No. 5.

Page Line Should Read 55 1 f b. 0<. < +0o 57 6 By lemma (xi) of e 61 4 o o condition (a) implies 5.o show that (a) is... 64 10 f. b immediate 74 4 f. b. o., C0 >, DQ > 0, with (I) non79 3 ~~79 3 y7'i > Vi(Gi)~ t 83 5 f b. /2k [f(t, xk(t), uk(t)) + 1]dt 92 10 f. b. and infinitely. o 118 9 xu +u 10 for u O, oo 121 2. o Linear in u 125 4 f. b... (1 - ak)uk] +... 132 5 f. b. conditions (b) and (d) hold. 133 7 f. b. from Chapter II, as is the case for A closed and contained in a slus as above. 6 f. b... A closed, but not contained.. 134 11 fb o o o A is closed, but not contained o. 8, 9, 10 f. b. Delete these lines. 135 9 conditions (b) and (d) holdO 138 4 f. b.... Liapunov o o

THE UNIVERSITY OF MICHIGAN SYSTEMS ENGINEERING LABORATORY Department of Electrical Engineering College of Engineering Technical Report SEL-67-24 EXISTENCE ANALYSIS FOR OPTIMAL CONTROL PROBLEMS WITH EXCEPTIONAL SETS by James Ro La Palm November 1967 This research was supported by NSF Grant GK-234o

ACKNOWLEDGMENTS The author wishes to express his gratitude to Professor Cesari for guidance and encouragement, and to thank the Systems Engineering Laboratory and the National Science Foundation for their support during the preparation of this thes is He wishes especially to acknowledge his wife's patience and understanding while he was preparing this thesis. This manuscript was typed by Miss Sharon Bauerle. ii

ABSTRACT The purpose of this thesis is to prove by direct methods existence theorems for optimal solutions in problems of optimal control and the calculus of variations. Indeed in each existence theorem we prove the existence of at least one element, in any given nonempty complete class 12 of admissible elements, which minimizes a given cost functional t2 lx, u] - f(t, x(t), u(t))dto t1 An admissible element is here a pair x(t), u(t), t < t < t2, of vector valued functions x(t) = (x,., x ), u(t) = (u,,u ), x(t) a trajectory and u(t) a control function, or strategy, for which the following requirements are made. (a) x(t) is absolutely continuous in [tl t2]; (b) u(t) is measurable in [t, t2]; (c) the pair x,u satisfies a given system of ordinary differential equations dx /dt = fi(t, x(t), u(t))1,il, 0 o, n, or dx/dt = f(t,x(t),u(t)), tl t < t2, in the sense of Caratheodory, f(t, x, u) = (f1, o 0, fn) being a given vector function; (d) x(t) satisfies a constraint on the time and space variables t and x of the form (t,x(t)) c A for all t e [tl, t2], where A is a given fixed subset of the tx-space E1 x En (e) u(t) satisfies a iii

As Cesari proved his Theorem I in 1966 by finally extending to Lagrange problems a Tonelli-Nagumo Theorem (1915-29) for free problems and n=l together with Filippov's statement (1959) for Pontryagin problems, so our Theorems II, III and IV extend to Lagrange problems analogous theorems of Tonelli for free problems and n=1. In Chapter III we deduce from Theorems II, III and IV as corollaries analogous existence theorems for problems with f linear in u. V

TABLE OF CONTENTS Page Introduction 1 Chapter L Statement of the Optimal Control Problem, Closure Theorems, and Existence Theorem I 10 1o Usual solutions. 10 2. Generalized solutions. 12 3o The distance function po 16 4o Upper semicontinuity of variable sets. 17 5. Properties (U) and (Q) of variable sets, 18 6. Closure theorem I, 24 7. Another closure theorem. 32 8. Notations for the optimal control problem. 39 9. Statement of existence theorem Io 41 l0. A few corollaries. 54 Chapter II Further Existence Theorems 58 11. An existence theorem with uniform growth of f o 58 12. An existence theorem with exceptional pointso 73 13. An existence theorem with a "slender" exceptional set. 92 Chapter III Optimal Control Problems Where f is Linear in u 121 14. A few lemmaso 121 1 5 Existence theorems where f is linear in u. 132 Bibliography 138 vi

LIST OF ILLUSTRATIONS Figure Page 1 An example where f lul 1 a+ oo as u -+ c, u e E1, pointwise in A, but not uniformly, 59 vii

INTRODUCTION The purpose of this thesis is to prove by direct methods existence theorems for optimal solutions in problems of optimal control and the calculus of variations. Indeed in each existence theorem we prove the existence of at least one element, in any given nonempty complete class Q of admissible elements, which minimizes a given cost functional t jx, u] -2 fot,x(t), u(t))dt. t An admissible element is here a pair x(t), u(t), tl < t < t2, of vector valued functions x(t) = (x o.. x ), u(t) = (u,o., u ), x(t) a trajectory and u(t) a control function, or strategy, for which the following requirements are made: (a) x(t) is absolutely continuous in [tl, t2]; (b) u(t) is measurable in [t1, t2]; (c) the pair x, u satisfies a given system of ordinary differential equations dxi/dt = f (t, xt), u(t)), i=l. n or dx/dt = f(t, x(t), u(t)), t < t < t2 in the sense of Caratheodory, f(t, x, u) = (f, o. o, f ) being a given si n vector function; (d) x(t) satisfies a constraint on the time and space variables t and x of the form (t, x(t)) ( A for all t [tL1, t2], where A is a given fixed subset of the tx-space E1 x E; (e) u(t) satisfies 1

2 a constraint of the form u(t) ( U(t, x(t)) for almost all t e [t1, t2], where U(t,x) is a given subset of the u-space Em and where U(t, x) may depend on both t and x; (f) the pair x, u is such that fo(t,x(t), u(t)) is L-integrable in [t, t2]o Usually, complete classes n of admissible pairs are obtained by considering all admissible pairs x(t), u(t), t1 < t < t2, whose trajectories x(t) satisfy given boundary conditions of a rather general type. The boundary conditions can be written in the McShane's form (tl, x(t1), t2, x(t2)) E B, where B is a given closed subset of the tlxlt2x2-space E2n+2O In Chapter I, for the convenience of the reader and for the necessary references, we repeat with variants an Existence Theorem I due to Cesari (1966), where a growth condition with respect to u on the scalar function f is assumedo Our results are contained in Chapters II and II In Chapter II we first prove an Existence Theorem II which is equivalent to Theorem I and in which the growth condition is expressed in terms of a uniform limit fo/ uj + oo as u - + oOo By means of counterexamples we show the need of the uniformity in our statement IL In the same Chapter II we prove then Existence Theorems III and IV, where we show that the growth condition can be relaxed in a closed exceptional subset E of A provided either an additional condition is satisfied at every point of E (Theorem 11) or no additional condition is satisfied but E is slender according to a suitable definition

3 (Theorem IV). As Cesari [4a] proved his Theorem I in 1966 by finally extending to Lagrange problems a Tonelli-Nagumo Theorem (1915-29) [17, 23a] for free problems and n=l together with Filippov's statement (1959) for Pontryagin problems [8], so our Theorems II, III and IV extend to Lagrange problems analogous theorems of Tonelli for free problems and n=l. Theorems II and IV which had been proved by Tonelli for free problems, n=l, and f of class C, had been extended by L. Turner [24] to free problems, any n > 1, and f of class C o Our extensions to Lagrange problems are also of 0 class C We owe to L. Turner the concept of slenderness we use in Theorem IVo In Chapter III we deduce from Theorems II, III and IV as corollaries analogous existence theorems for problems with f linear in Uo As in Cesari's work we assume A to be closed, and U(t, x) also closed for every (t, x) E A (though not necessarily compact) and satisfying Kuratowski's condition of upper semicontinuity (property (U))o Analogously, for the sets Q(t, x) and Q(t, x), which are the images of U(t, x) under f and f - (fo, f), and for the derived sets Q (t, x), we assume with Cesari that the modified Kuratowski upper semicontinuity condition for convex sets is satisfied (property (Q)). As in Tonellivs and Cesarivs works we first prove the existence theorems for A compact, and then we extend them to the

4 case of A closed under usual additional assumptions. Our proof of Theorem II in Chapter II is given by means of five lemmas. By these lemmas we prove first the equivalence of the conditions in Theorems I and II, and then the statement that the uniformity requested in our Theorem II need not be explicitly verified for free problems, since for these problems the uniformity is a consequence of the remaining hypotheses. This last statement is relevant since in the corresponding Tonelli's Theorem II such a uniformity was not requested. In such a way we can conclude that our Theorem II contains as a particular case the corresponding Tonelli statement for free problems and n=1, as well as Turner's statement for free problems and any n > 1. In order to describe our proofs of Theorems III and IV we must first summarize Cesari's scheme for the proof of Theorem I. This proof of Cesari is a modification of the usual direct method in the calculus of variations for free problems, in the sense that an application of Helly's Theorem and suitable modifications of a closure theorem due to Ao F. Filippov replace Tonelli's lower semicontinuity argument. We describe Cesari's proof only for the case A compact. The first step is to show that the infinium i of 1[x, u] is finite, where the infimum is takern over a11 pairs x, u of the given complete class C of admissible pairs. Thlus, there exists a minimizing

5 sequence xk(t) uk(t), tlk < t t2k, of pairs in, that is, a sequence such that I[xk Uk] - i as k - + oo. The second step consists in showing that the trajectories xk(t), tlk < t < t2k' k=l, 2, o., of such a minimizing sequence are equiabsolutely continuous and also equibounded. Then, AscoliYs Theorem guarantees the existence of some subsequence of integers k and of some continuous vector function x(t), t < t < t2, such that tlk t, tk -t2 and xk(t) x(t) in an obvious modification of the uniform topology as k - + o along the extracted subsequenceO The equiabsolute continuity of the trajectories xk(t), tlk < t < t2k9 kl 2, o..., and the uniform convergence xk -x guarantee then that x(t) is also absolutely continuous in [t1, t2]. The third step concerns the sequence xk (t), tk < t < t2k< k=1 2, o.., with t t xk(t) vk(t)dt fo(T Xk(T), Uk ()d tlk tlk for which no equicontinuity can be proved at this point, neither under the conditions of Theorem I, nor under the conditions of Theorems II, III and IV. Instead of the functions xk(t), the two functions t t Yk(t) = - vk(t)dt, Ykt) - kt)dt tlk < t < t2k' tk lk are taken into consideration, where as usual

6 v+ kl, vk - kl V kik/2 Vk (Vk +V V J k and hence + ~- +I + - vk vk >, vk Vk Vk9 ii k + Vk and x (t) Y kt)+ Yt), lk< t<t k 1,2 9 Now the functions Yk(t) are monotone nonincreasing and uniformly Lipschitzian, and the functions Yk(t) are nonnegative, monotone nondecreasing and uniformly bounded. By applying Helly's Theorem to the sequence Yk(t) and then AscoltVs Theorem to Ykit), further suck k~t) I cessive extractions are obtained such that xk(t) Yk(t) + Yk(t), tk t < t2k, k=1,2o.., converges for eacht, t t < t < t2, toward x (t) = Y(t) + Z(t), where Y(t) is a (scalar) absolutely continuous function, and Z(t) is a nonnegative monotone nondecreasing function with ZI(t) = 0 almost everywhere in [t1 t, and Y(t1) = Z(t1) = 0 Since Jk 2kt + Yk2k " ^^2^ "o ^\ -i as k-+oo, k6+2k) Yk t2k)= xkt2k) = [Xk k k+ then by taking the limit along the last extracted subsequence one obtains Y(t2) + Z(t2) = i where Z(t2) > 0, and hence Y(t2) < i. The fourth step consists in applying a closure theorem which is a generalization by Cesari of one due to Ao F. Filippovo This guarantees the existence of a measurable control u(t), t1 < t < t2, such that x(t), u(t), t1 < t < t2, is an admissible pair for the problem, and I[xu] Y(to) < io

7 The fifth and final step consists in a simple application of the completeness property of S~ in order to conclude that the pair x, u belong to the class 2 and hence i < l[x, UJo Therefore, I[x, u] = i and the proof is complete, in the case A is compact. The proof of Existence Theorem III in Chapter II repeats essentially the same steps, but the proof that the trajectories xk(t), tlk < t <( t2k9 k =l 2..., are equiabsolutely continuous is much more complicated based as it is on the growth property at the points not on the exceptional set E and on the assumed additional property at the points of E. This part of the proof is inspired by the corresonnding part of the analogous theorem of Tonelli for free problems. The proof of Existence Theorem IV in Chapter I also differs from Cesari's proof in step two. First the trajectories xk(t), tk < t < t2k9 k=l 2,0.. are proved to be equicontinuous and of uniform bounded variation. Thus the limit vector function given by Ascoli's Theorem is continuous and of bounded variation. Finally a proof by contradiction shows that x(t) is also an absolutely continuous vector function. The other steps in the proof are essentially the same as in Cesari's proof. The long and difficult argument replacing step two again is modeled on the corresponding part of the analogous argument of L. Turner for free problems. Theorems I to IV extend the analogous theorems of Tonelli for free problems with the exception of Theorem IV where an

8 additional hypothesis is being made. Although the present work concerns only usual solutions, we can say that existence theorems for generalized solutions can easily be derived from Theorems I to IV. Indeed, as pointed out by R. V. Gamkrelidze [9], the generalized solutions of a given problem can be thought of as usual solutions of an analogous problem obtained by a suitable relaxation of the given problem and with an enlarged set of control variables. Thus, as shown by Cesari [4b] for Theorem I, also Theorems II, III and IV yield analogous existence theorems for weak solutions. Generalized solutions, which had been introduced for free problems by L. Co Young [27], have been studied extensively for free and for Lagrange problems by E. J. McShane [16], Jo Warga [25], A. Plis [19], Ro V. Gamkrelidze [9], and Cesari [4b]o There are various other approaches for obtaining existence theorems for problems of optimal control and the calculus of variations. A few of these are briefly described below. Nonlinear existence theorems have been proven by E. 0. Roxin [22] who employed the concept of the attainable set and also by Eo Lee and L. Markus [14]. Lo Neustadt [18] has proven an existence theorem for the Pontryagin problem, where the functions f and f are assumed to be linear in the state x and U is a fixed compact subset of the uspace Emo In an existarce theorem due to Lo W. Neustadt no convexity

9 condition is required. For the proof he employs a theorem by A. Lyapunov on the convexity of the range of a vector measure, which was previously used by H. Halkin [10] to derive necessary conditions for optimal control problems, and the concept of the attainable set, which was previously used by E. Roxin [22]. Do Blackwell [2], Ho Chernoff [5], H. Hermes [12] and P. R. Halmos [11] have given alternate proofs or extensions of the theorem due to A, Lyapunov on the convexity of the range of a vector measure. Ao V. Balakrishnan [1] has treated optimal control problems in abstract function spaces. E. H. Rothe [21] has proven an existence theorem for multidimensional free problems of the calculus of variation by utilizing Sobolev's imbedding theorems. For problems of optimal control with U(t) depending on t only, L. S. Pontryagin [20] gave his now famous principle of maximum as a wide ranging necessary condition. This principle has been extended in many ways by R. V. Gamkrelidze, Ho Halkin, L. Neustadt, and others.

Chapter I Statement of the Optimal Control Problem, Closure Theorems, and Existence Theorem I 1. Usual solutions. Let A be a closed subset of the tx-space E1 x En, t e E1, 1 Kn x = (x, o o x ) E En, and for each (t, x) E A, let U(t,x) be a closed subset of the u-space E, u=(u,, um). We do not exclude that A coincides with the whole tx-space and that U coincides with the whole u-space. Let M denote the set of all (t, x, u) with (t, x) e A, u e U(t, x). Let f(t,x, u) = (fo, f) = (fo, f1 "'' f ) be a continuous vector function from M into En+o Let B be a closed subset of points i n 1 n (t1,xlt2,x2) ofE2n+29 x1 = (X,,X1), x2 =(x2,..,x2). We shall consider the class of all pairs x(t), u(t), tl < t < t2, of vector functions x(t), u(t) satisfying the following conditions: (a) x(t) is absolutely continuous (AC) in [tl, t2], (b) u(t) is measurable in [tl, t2]; (c) (t,x(t)) E A for everyt e [t1,t2]1 (d) u(t) e U(t,x(t)) almost everywhere (a. e. ) in [tl, t2]; (e) fo(t, x(t), u(t)) is L-integrable in [t1 t2]; (f) dx/dt = f(t, x(t), u(t)) a. e. in [t1, t2], (g) (tlx(t1 ), t2,x(t2 ) c Bo 10

11 By (f) we mean that the n ordinary differential equations dxi/dt = f.(t,x(t),u(t)), i=1,2 0n, (1) are satisfied a. e. in [t1, t2]. Since x(t) is AC, that is, each component x (t) of x(t) is AC, we conclude that all f.(t,x(t),u(t)), i=l,2,.oo,o n, are L-integrable in [tl,t2] as fo' A pair x(t), u(t) satisfying (abcdefg) is said to be admissible and for such a pair x(t) is called a trajectory and u(t), a strategy, control, or steering function. As usual, U(t, x) is said to be the control space at the time t and space point x. The functional t2 I[x,u]= f fo(t, x(t), u(t))dt (2) t1 is called the cost functional, and we seek the minimum of I[x, u] in the total class 2 of admissible pairs x(t), u(t), or in some well defined subclass of f In the particular case where U(t, x) is a compact subset of Em for every (t, x) E A, the problem of the minimum of I[x, u] is called a Pontryagin problem of optimal control theoryo The general case above, where U(t, x) is a closed but not necessarily compact subset of Em for every (t, x) e A will be denoted as a Lagrange problem with unilateral constraints or as the optimal control problem. The classical Lagrange problem corresponds essentially to the case where

12 U = Em is the whole u-space, with the side conditions being here differential equations in normal form. In the particular case in which fo0=1 then I[x, u] = t2 - tl and the problem of minimization under consideration is then called a problem of minimum transfer time (from the state x(tl) to the state x(t2)). There is another particular case of the Lagrange problem which shall be taken into consideration, namely m=n, U=Em and the vector function f(t, x, u) given by f(t, x, u) = u, or f.(t, x, u) - u, i=1,., m=n, and hence f(t, x, u) = (f, u). Then the differential system (1) reduces to dxi/dt = u i, -, o. n, and the cost functional becomes t2 I[x] f f(t, x(t), x(t))dt (3) t1 This problem is called a free problem. 20 Generalized solutions. Often a given problem has no optimal solution, but the mathematical problem and the corresponding concept of solution can be modified in such a way that an optimal solution exists and yet neither the system of trajectories, nor the corresponding values of the cost functional are essentially modified. The modified (or generalized)

13 problem and its solutions are of interest in themselves, and have relevant physical interpretations. Essentially, we consider a finite system of distinct strategies which are thought of as being used at the same time according to some probability distributiono Instead of considering the usual cost functional, differential equations and constraints t2 ti lx,u] S fo(t, x(t), u(t))dt, dx/dt f(t, x(t), u(t)), f - (fl,. 0 f), (4) (t, x(t)) A, u(t), U(t,x(t)), we consider a new cost functional, differential equations and constraints t2 J(x,, ) - S g(t, x(t), p(t), v(t))dt, t1 dx/dt = g(t,x(t), p(t), v(t)), g (g, g) (5) (t, x(t)) E A, v(t) E V(t, x(t)), p(t) e r. Precisely, v(t) = (u() 0. u ) represents a finite system of y > rn+l ordinary strategies u9 o. 9 u, each u having its values in Ut, x(t)) c E. Thus, we think of v (u o.. u ) as a vector variable whose y components u1..,u are themselves vectors

14 u with values in U(t, x). In other words v = u(1o),,u) u(J E Ut, x), j=1,o y, or ve V(t,x) = [U(tx)] Ux... XUc Emy, (6) where the last term is the product space of U by itself taken y times, and thus V is a subset of the Euclidean space Em o In (5) p = (P,...,p ) represents a probability distribution. Hence, p is an element of the simplex r of the Euclidean space E defined by p> 0, P1 +'* + P = 1. Finally, in (5) the new control variable is (p,v), with values (p,v) E F x V(t,x) cE + In (5) g = (g1.o gn), and all gog, g o.o,g are defined by gi(t, x, p, v) = Pjfit, x^ U(J) i=o 1, o. n. (7) j=1 As usual we shall require that the functions p(t), v(t), t1 < t < t2, are measurable and that x(t), t1 < t < t2, is absolutely continuous. As in No. 1, we shall require as usual that x(t) satisfies boundary conditions of the type (tl, x(tl), t2 x(t2)) ( B c E2n+2 where B is a given closed subset of E2n+2o As in No. 1, we require go(t, x(t), p(t), v(t)) to be L-integrable in [tl, t2] We shall say that [p(t),v(t)] is a generalized strategy, that p(t) = (P1, o o, ) is a probability distribution and that

15 v(tr = (u (1) u /)) is a finite system of (ordinary) strategies. We shall say that x(t) is a generalized trajectory. It is important to note that any (ordinary) strategy u(t) and corresponding (ordinary) trajectory x(t) (thus, satisfying (1)) can be interpreted as a generalized strategy and generalized trajectory, by taking v(t) (uUJ)(t), j=l,...,y) and p(t) - (p(t), j=l,...,y) defined by u(i)t) = u(t), p. (t) l/y, jl,....,y Then relations (5) reduce to relations (1). Instead of the usual set M we shall now consider the set N c E1 +,+m of all (t, x, p, v) with (t, x) e A, p e r, v e V(t,x). As usual, we shall assume that A is a closed subset of E1 x En, and that f - (f0, f,. o O 0 f0 ) is a continuous vector function from M into E n~l Under hypotheses which are often satisfied, any generalized trajectory can be approached as closely as we wish by means of usual solutions9 and correspondingly the value of the cost J[x, p, v] can be approached as closely as we wish by the value of the usual cost l[x, ul In this sense we shall understand that the usual solutions and the corresponding values of the cost functional are not essentially modified by the introduction of generalized solutions. The existence theorems of the present thesis apply to generalized as well as to usual solutions provided go and g are replaced for f and fo More details concerning the application of Cesari's Existence

16 Theorem I to generalized solutions are given in [4b]. 3o The distance function p. If we denote by X the space of all continuous vector functions 1 n x(t) = (x 9.. o, x ), a t _ b, from arbitrary finite intervals [a.,b] to En it is convenient to define a distance function p(x, y) for elements x(t), a' t b, and y(t), c < t_ d, of X, so as to make X a metric space. For this purpose we extend x(t) in all (-oo, +oo) by defining it equal to x(a) for t - a and equal to x(b) for t i b, andx analogously, for y(t). We then define p(x,y) = la-c i+ jb-dj + max Ix(t)-y(t) I, where the maximum is taken for all t, -oo < t < +oo Then p is a distance function and X is a metric space. Given functions xk(t), ak < t < bk, k=1,2,. o o and x(t), a < t < b, we shall say therefore that xk x as k - o in the p-metric if p(xk, x) - 0 as k - oo. If the interval [a, b] is fixed, then this reduces to the usual uniform convergence. For any admissible pair [x(t), u(t)] the trajectory x(t) is an element of X, but of course an element of X may not be the trajectory of an admissible pair. A class 2 of admissible pairs is said to be complete provided it satisfies the following property: If xk(t), uk(t), tlk -' t - t2k k=, 2,. o., and x(t), u(t), t< t < t2 are all admissible pairs, if k~~~~~~l,~~~~2,'.,adtu 1==2

17 xk(t) - xt) as k oo in the p-metric, and if all pairs xk(t), uk(t), k=l, 2,..., belong to a then x(t), u(t) also belongs to S The classes usually taken into consideration in applications are completeo The class of all admissible pairs (satisfying (abcdefg)) is certainly complete. 4. Upper semicontinuity of variable sets. Using the same notations of No. 1 we shall denote by Q(t, x) the set Q(t,x) - f(t,x, Ut,x)) = [z En Z =f(tx, u), u U(t, x)] c En Here Q(t, x) is the image in En of the set U(t, x) in the mapping U(t, x) - En defined by z = f(t, x, u), u e U(t, x). If f is continuous, as assumed in No0 1 and if U(t, x) is compact for every (t,x) e A, then also Q(t, x) is compact. Given any point (to x ) e A and 6 > 0, we denote by N (t, x ) the set of all (t, x) A A at a distance < 6 from (to, xo) and denote by N0O(t x ) those points of A at a distance < 6 from (t0, x )o The set U(t, x) is said to be an upper semicontinuous function of (t, x) in A provided for every (t, X) ) A there is a 6 > 0 such that U(t,x) c [U(to,)] 0 Xo)

18 for every (t, x) E Ne (t Xe), where U denotes the closed E -neighborhood of U in E If U(t, x) is compact for every (t, x) e A and an upper semicontinuous function of (t, x) in the closed set A, then Cesari [4a] has proven that Q(t, x) is also compact for every (t, x) e A and an upper semicontinuous function of (t,x) in A. 5. Properties (U) and (Q) of variable sets. If E denotes any subset of En we shall denote by cl E the closure of E and by co E the convex hull of Eo Thus cl co E denotes the closure of the convex hull of E, or briefly, the closed convex hull of E. Let U(t,x), (tx) e A, be a variable subset of E For every 6 > 0, let U(t,x,6 ) - UU(t',x'), where the union is taken for all (t',x') e N6 (t,x). We shall say that U(t,x) has property (U) at (t,x) e A if U(gtx) n clU(t,x,6) n c= u(t,x). 6 >0 6 >0 (t,x)eN6 (t,x) We shall say that U(t,x), (t, x) A, has property (U) in A if U(t,x) has property (U) at every point (t, x) E Ao If a set U(t, x) has property (U), say at (tx), then obviously U(t, x) is closed for it is the intersection of closed sets, Let Q(t, x), (t,x) e A, be a variable subset of Eno For every

19 (t, x) e A and 5 > 0, let Q(t,x, 6) = UQ(t, x') where the union is taken for all (tI,x ) e N6 (t,x). We shall say that Q(t,x) has property (Q) at (t,x) c A if Q(t,x) =n clcoQ(f,x, 6) n clco U Q(t,x). 6 >0 6 >0 (t, x)E N (t,x) We shall say that Q(t, x) has property (Q) in A if Q(t, x) has property (Q) at every point (t, x) E A. If a set Q(t, x) has property (Q), say at (f, x), then obviously Q(t, x) is closed and convex, as the intersection of closed and convex sets. Property (U) is Kuratowski's concept of upper semicontinuity [13] used also by Choquet [6] and Michael [15]. The following statements (i) - (viii) and their proofs are given in [4a]. (i) If A is closed and U(t, x) is any variable set which is a function of (t, x) in A and has property (U) in A, then the set M of all (t, x u) A x Em with u e U(t,x), (t,x) E A is closedo (ii) If the set U(t x) is closed for each (t x) e A and is an upper semicontinuous function of (t, x) in A, then U(t, x) has property (U) in Ao Thus, for closed sets the upper semicontinuity property implies property (U) but the converse is not true, that is, the upper semicontinuity property for closed sets in more restrictive than property (U)o This is shown by an example in [4a].

20 (iii) If A is compact, if U(t, x) is compact for every (t,x) e A and is an upper semicontinuous function of (t, x) in A, then M is compacto (iv) Property (Q) at some (t, x) implies property (U) at the same (t,x), and U(t,x) n= n clcoU(f,x, 6) = clU(f,x, 6) 6 >0 6 >0 n u(t,x,G)o 6 >0 Analogously, if U(t,x) has property (U) at (t,x), then U(t,x) = clU(tx, ) n U(t,x,6 ). 6>0 6<0 (v) If for each (t, x) E A the set U(t, x) is closed and convex, and U(t, x) is an upper semicontinuous function of (t, x) in A, then U(t, x) has property (Q) in Ao Let us now consider the sets Q(t, x) f(t, x, U(t, x)), (t,x) e A, Q(t, x) c En, which are the images of sets U(t, x) c E, for each (t, x) e A. The hypothesis that A is compact, that f is continuous on M, that U(t, x) has property (Q) [or (U)] in A, and that Q(t, x) is convex for each (t, x) e A does not imply that Q(t, x) has property (Q) [or (U)] in A. Even the stronger hypothesis that A is compact, that f is continuous on M, that U(t, x) has property (Q) in A, and that Q(t, x) is compact and convex for each (t, x) E A, does not imply

21 that Q(t, x) has property (Q) in Ao These statements are shown by two examples in [4a]o However, the following statement is valid. (vi) If A is closed and f is continuous on M, if U(t, x) is compact for each (t, x) E A and U(t, x) is an upper semicontinuous function of (t, x) in A, then Q(t, x) possesses the same property, and also has property (U) in A. If we know that Q(t, x) is convex, then Q(t, x) also has property (U) in A. If we know that Q(t, x) is convex, then Q(t, x) also has property (Q) in A. REMARK: The statements above show that properties (U) and (Q) are generalizations of the concept of upper semicontinuity for closed, or closed and convex sets, respectively. (vii) If A is a closed subset of the tx-space E1 x En, if U(t, x), (t, x) e A, U(t, x) c Em is a variable subset of Em satisfying property (U) in A, if M denotes the set of all (t, x, u) with (t,x) e A, u E U(t,x), if fo(t,x, u) is a continuous scalar function from M into the reals, if U(t, x) denotes the variable subset of Emi defined by U = [u = (u u) e E m+ u > f (t,x,u), u e U(t,x)], then U(t, x) satisfies property (U) in A. An example is given in [4a] which shows that U(t,x) of statement (vii) does not necessarily have property (Q) in A even if we assume that U(t, x) has property (Q) in A and fo(t, x, u) is convex in u for each (t, x) e Ao

22 However a slightly stronger statement does hold. It is necessary to introduce a new property for this statement. A scalar function fo(t, x, u) (t, x, u) e M is said to be quasinormally convex in u at (to, x, u0') E M provided, given e > 09 there are a number 6 = 6 (to9 Xo0 uo9 r) > 0 and a linear scalar function z(u) =r+bo u, b=(bl..o,bm), r, bI,. o.,bm real such that (a) fo(t,x,u) > z(u) for all (t,x) r N (tO x), u E U(t,x) (b) fo(t, x, u) < z(u) + e for all (t, x) E N6 (to Xo)9 ue U(t,x), lu - uo I 60 The scalar function f (t x, u) is said to be quasi-normally convex in u, if it has this property at each (t, x, u ) e Mo (viii) If A is a closed subset of the tx-space E1 x En, if U(t,x), (t,x) ( A, U(t,x) c Em, is a variable subset of E satisfying property (Q) in A, if M denotes the set of all (t, x, u) with (t, x) e A, u e U(t,x), if fo is a continuous scalar function on M, which is convex in u for each (t, x) (e A, if either (a) the sets U(t, x) are all contained in a fixed solid sphere S of Em, or (p) the function f (t, x, u) is quasi-normally convex in u at every (t, x, u) e M, then the set U(t, x) of statement (vii) has property (Q) in Ao A function fo(t, x u) is said to be normally convex in u at (to, X, u) e M, if for each (to x u) e M and for every e > O there are constants 6 =(t^,x ^u,u) E> 0, v =V(t^ x,u^9 E) > O and a

23 function z(u) c + do u such that (a) fo(t,x, u) > z(u) + vl u-uo, for each (t,x) E N (to, x), u e U(t,x) and (b) f(t, x, u) < z(u) + e for each (t,x) e N6 (to, X), u eU(tx), u-uo I < 6. The scalar function f (t, x, u) is said to be normally convex in u, if it has this property for each (to x, u ) e M. We shall need the following statement due to Tonelli [3aa] and L. Turner [24]. (viii) f (t x, u) is normally convex in A x Em if and only if f (t, x, u) is a convex function of u for each (t, x) e A, and for no points (to Xo) E A, u,u1 E Em with ju1 1 0 it is true that for all real X fo(t x u +{f ~+fo t xt u -Xu) = fo(to, X x U) (8) 0 0o 0 I 0\ 09 1 O'0 0' 0 (ix) If (t, x, u) is a convex function of u for each (t, x) e A and fo(t,x, u) ju j1 + oo as Ju +oo for each (t,x) e A, then f(t, x, u) is normally convex in A x Em PROOF: By statement (ix), it suffices to prove that there exists no points (t, x ) E A, uo U1 e E with |ul I j 0, for which 00 0, u m Ju relation (8) holds. Suppose such points exist. Then, 2 {f (to x, u0 + Xu) - f(t,x, u Xu )} I u + Xu1 1 = f (t9 x0, u ) + X -1

24 when u + Xu O. Now f (to, xo, uo) u + XU1 -1 as X -+ oo. As f(t, x, u) Iu I -+oo as ju| -+oo for each (t, x) e A, this statement implies that fo(to, X,u + Aul) u as +Xu a Thus, f (to x,u - Xu1) U + Xu -1 = f(t,xu - Xul) ~ lju - Xu1 Ij (1u - Aul I lUo + Au1l ) -+oo as - +oo. This is a contradiction and the statement is proven. (x) If fo(t, x, u) is a convex function of u for each (t, x) e A, and there exists a function 4(z), 0 < z < +oo, such that f(t, x, u) > 4 Iu I) for each (t, x, u) E A x E and 4(z)z -+oo as z - +oo, then f (t, x, u) is normally convex in 0 AxE m PROOF: One has fo(t,x,u) u -1 > (luI) JIl for u i0 and (t, x, u) e A x Em. As 4(z)z -+oo and z - +oo, therefore fo(t,x, u) juj - +oo as u| - +oo. Statement (ix) applies and this statement is proven. 6. Closure Theorem I (Cesari [4a]). Let A be a closed subset of E1 x En, let U(t, x) be a closed subset of Em for every (t, x) e A, let f(t, x, u) = (fl... f ) be a continuous vector function on the set M = {(t, x, u) I (t, x) e A, u e U(t, x)} into En, and let

25 Q(t, x) = f(t, x, U(t, x)) be a convex subset of E for every (t, x) e A. Assume that U(t, x) has property (U) in A, and that Q(t, x) has property (Q) in Ao Let xk(t), tkk < t < t2k k=l 2,, be a sequence of trajectories, which is convergent in the metric p toward an absolutely continuous function x(t), tL < t < t2o Then x(t) is a trajectory. REMARK: If we assume that U(t, x) is compact for every (t, x) E A, and that U(t, x) is an upper semicontinuous function of (t, x) in A, then the set Q(t, x) has the same property, U(t, x) has property (U), Q(t, x) has property (Q), and Closure Theorem I reduces to one of A. Fo Flipppov [8] (not explicitly stated in [8] but contained in the proof of his existence theorem for the Pontryagin problem with U(t, x) always compact). PROOF: The vector functions tt) = xI t)), t < t < t2,,k(t) - Xk(t) x f(t, ), uk(t))3 tk t < t k=l,2,. o, (9) are defined almost everywhere and are L-integrable. We have to prove that (t, x(t)) e A for every t1 < t < t2, and that there is a measurable control function u(t), t1 < t < t2, such that (t) xg (t) = -g(t, x() (t))t ut) E Ut x(t)) (10) for almost all t e [t~, to],

26 First, p(xk, x) - as k -o, hence tlk - tl, t t2. If t e (t, t2) or t < t < t2, then tlk < t < t2k for all k sufficiently large and (t, xk(t)) e A. Since xk(t) - x(t) as k- oo and A is closed, we conclude that (t,x(t)) e A for every tl < t < t2. Since x(t) is continuous, and hence continuous at tl and t2, we conclude that (t, x(t)) e A for every tl < t < t2. For almost all t e [tI, t2] the derivative x'(t) exists and is finite. Let to be such a point with tl < t <.t2. Then there is a a > O with tl < t -< t + a < t2, and, for some k and all k > ko, also tlk < t0 - a < to + a < t2ko Let xo 0 x(to). We have xk(t) - x(t) uniformly in [t - a, to + a] and all functions x(t),xk(t) are continuous in the same interval. Thus, they are equicontinuous in [to - a, to+ a]. Given e > 0, there is a 6 > 0 such that t, t' [t - a, t + a], t-t' <, k > k, imply jx(t) - x(t') i e/2, xk(t) - xk(t') < e/2. We can assume 0 < 6 < a, 6 < o For any h, O < h < 6, let us consider the averages h mhk = h- f k(to+s)ds = h l[xk(to+h) - xk(to)] O

27 Given r > 0 arbitrary, we can fix h, 0 < h < 6 < c, so small that Imh - (to)I < r. (12) Having so fixed h, let us take kl > ko so large that Imhk - mh <, 1xk(to) - x(to) I E/2 (13) for all k > k1. This is possible since xk(t) - x(t) as k - oo both at t = to and t = t + h Finally, for 0 < s < h, k > kl, Ixk(tos) - xto) I< lxk(to+S) - xk(t) + lxk(to) - (to) < e /2+ e /2 = e, l(t0 +s) -t0 < h < 6 < and f(to + s, xkto + s)g Uk(to + s)) E Q(to o s+ s)). Hence, by the definition of Q(t, xo, 2E), also k (to+ ) = fto + s xk k(to + s )) (t+o, o, 2E. The second integral relation (11) shows that we have also mhk e cl co Q(t, xo,9 2e), since the latter is a closed convex set. Finally by relations (12) and (13), we deduce 1(to) - mhk < |(to) - mh + Imh- mhk I 2, and hence

28 )(to) e [cl co Q(to, xo, 2e )]2c ~ Here 7 > 0 is an arbitrary number, and the set in brackets is closed. Hence, 4(to) E cl co Q(to, X, 2e), and this relation holds for every e > 0. By property (Q) we have (to) e Ch cl co Q(t, xo;2e) = Q(to, xO) where xo = x(t), and Q(t, x ) = f(to xo, U(t, x))0 This relation implies that there are points u = u(to) E U(to, x) such that b(to) = f(to, (t ), u(to)) (14) This holds for almost all to E [t1, t2], that is, for all t of a measurable set I c [t, t2] with meas I = t2 - t1 If we take Io = [tl,t2] - I, then meas Io = 0. Hence, there is at least one function u(t), defined almost everywhere in [tl, t2], for which relation (14) holds a. e. in [tl, t2]o We have to prove that there is at least one such function which is measurable. For every t e I, let P(t) denote the set P(t) = [uuE U(t,x(t)), ~(t) = f(t,x(t),u)] c U(t,x(t))c Em We have proved that P(t) is not empty. For every integer X = 1, 2, o, there is a closed subset C of I, Cx c I [tlt2], with meas C > max [0, t2-t - 1/X], such that f(t) is continuous on C2. Let WA be the set

29 Wx: [(t, ) t ECA ue P(t)] c E1 E Let us prove that the set Wx is closed. Indeed, if (t, u) is a point of accumulation of WX, then there is a sequence (t, u), s=l, o.., with (ts,) U Wxs ts t u - u. Then, t e Ch and t e CC since Ck is closed. Also x(t ) -x(t), k(t ) -~(t), and since (t, x(ts))e A, (t) f(t xt ), u(t)), (t x(ts ) u(t ))e M, we have also (t, xt()) E A, (t,x(t),u) E M, because A and M are closed, and 0(t) = f(t x(t), u) because f is continuous. Thus, u E P(t), and (t,x) e W. This proves that W is a closed set. For every integer f let W.; P&(t) be the sets W = - [(t, u) I (t u) E WX ul< ] c WX c Elx E P(t) - [ul u e P(t), ul < ] c P(t) c U(t, x(t)) c E, Ca X [tl (t, u) ( W. for some u] c C I c [tl, t2]. Obviously, WX( is compact2 and so is CXf as its projection on the t-axis. Also U).C - CXC and WX is the set of all (t, u) with t e Cxf, u E PfQt). Thus, fort E CX, Pf(t) is a compact subset of U9t, x(t)). For t e Cxf and f large enough, the set P (t) is a nonempty compact subset of all u = u,. o u )E U(tx x(t)) with f(t,x(t), u) = 0(t) and J u <. Let P be the subset of P with u minimum, let P2 be the subset of P1 with u minimum,..., let Pm be the subset of Pmi with u mirimuvm. Then Pm is a single Pm m-l m rn

30 point u - u(t) E U(t, x(t)) with u(t) = (u, m)t e Cm lu(t)l <, and f(t, x(t), u(t)) = p(t)o Let us prove that u(t), t E CX, is measurable. We shall prove this by induction on the coordinates. 1 s-i Let us assume that u (t),...,. u (t) have been proved to be measurable on C^ and let us prove that u (t) is measurable. For s=l nothing is assumed, and the argument below proves that u (t) is measurable. For every integer j there are closed subsets CX.j of CX withC C CX; j+' meas CXj > max[0, meas CX-l/j], I s-l such thatui (t)... a u (t) are continuous on C j.o The function p(t) is already continuous on C and hence q(t) is continuous on every set CX and C jo Let us prove that u (t) is measurable on Cj We have only to prove that, for every real a, the set of all t e Cj with u (t) < a is closedo Suppose that this is not the case. Then there is a sequence of points tk e CXj with u (tk) < a, tk -t e C j, u (t) > a. Then (tk) -(f), u (tk) -ua(t) as k-oo, a - 1,...,s-lo Since I u1(tk) I < for all k and,=s, s+l,. oo, m, we can select a subsequence, say still [tk], such that u(tk) -u as k -oo, -=s, s+1, o, m, for some real numbers u. Then tk -t, x(tk) -x(t), u(tk) -u, where ~1 - s-1 ~s -m u - (u (t),..o,u (u e. u ). Then, given any number r7 > 0, we have u(t,) c U(tk, x(tk) c cl U(t, x(t),,7)

31 for all k sufficiently large, and, as k -oo, also u ecl U(t, x(t), r/). By property (U) we have u n clU(t,x(tr) = U(t,x(f)). > O On the other hand p(tk) f(tk x(tk), u(tk)), u (tk) < a, yield as k -oo, ~(t) f(t,x(t),u), u < a, (15) while t e CXQ implies 0(t) - f(t,x(t),u(t)), u(t) > a. (16) Relations (15) and (16) are contradictory, because of the minimum property with which u (t) has been chosen. Thus, u (t) 5 - is measurable on C.j for every j, and then u (t) is also measu- rable on Cio By induction, all components u (t),.., u (t) of u(t) are measurable on CA, hence, u(t) is measurable on CX. Since UC = - C, meas C. > meas I - V/, we conclude that u(t)) is measurable on every set C and hence on I, with meas I = t2 - t1. Thus u(t) is defined a. e. on [t1, t2], u(t) e U(t, x(t) and f(t, x(t) u(t) ) = (t) a. e. on [t1l t2]. Closure Theorem I is thereby proved. REMARK: The last part of the proof of Closure Theorem I

32 concerning the existence of at least one measurable function u(t) is a modification, for U(t, x) closed and satisfying property (U), of the analogous argument of A. Fo Filippov [8] for the case where U(t, x) is an upper semicontinuous compact subset of the Euclidean space Em. A different argument - again concerning only the last part of the proof - has been devised by C. Castaing:[3]. His result concerns a multi-valued map F: t - I1t), with I(t) depending on t only. If F(t) - U(t, x(t)) and U(t) U(t, x(t)) is an upper semicontinuous function of the time t, then Castaing's result provides a different argument for the second part of the closure theorem. 7o Another closure theorem s Let us denote by y - (x, o. 0 x ) the s-vector made up of cer1 s 1 n tain components, say x,o. 0, O < s< n, of x - (x,o0 x ), and by z the complementary (n-s)-vector z (xS, + o x) of x, so that x = (y, z). Let us assume that f(t, y, u) depends only on the coordi1 s nates x, o,x of Xo If x(t), t < t < t29 is any vector function, we shall denote by x(t) = [y(t), z(t)] the corresponding decomposition of x(t) in its coordinates y(t) - (x, o x ) and z(t) = (xS+ o o ox) I s We shall denote by A a closed subset of points (t, x x ), that is, a closed subset of the ty-space E x Es, and let 1 s A A x E o Thus, A is a closed subset of the tx-space o n-s E XEn

33 Closure Theorem IL (Cesari [4a], ) Let A be a closed subset of the ty-space E1 X Es and then A = A En is a closed subset of l i s 0 n-s the tx-space E x E. Let U(t, y) denote a closed subset of E for every (t, y) e A o let M be the set of all (t, y, u) E El+s with (t, y) e Ao, u e U(t, y), and let f(t, y, u) = (f lo fn) be a continuous vector function from M into En. Let Q(t, y) = f(t, y, U(t, y)) be a closed convex subset of E for every (t, y) e A. Assume that U(t, y) has property (U) in A and that Q(t, y) has property (Q) in Aoo Let xk(t), tlk < t < t2k2 k=1, 2. o, be a sequence of trajectories, xk(t) = (Yk(t), zk(t)), for which we assume that the s-vector Yk(t) converges in the p-metric toward an AC vector function y(t), tl < t < t2, and that the (n-k)-vector zk(t) converges (pointwise) for almost all tl < t < t2, toward a vector z(t) which admits of a decomposition z(t) = Z(t) + S(t) where Z(t) is an AC vector function in [t1, t2], and S'(t) 0 a. e. in [tl, t2] (that is, S(t) is a singular function). Then, the AC vector X(t) - [y(t), Z(t)], tl < t < t2, is a trajectoryO REMARK: For s=n, this theorem reduces toClosure Theorem I, PROOF: The vector functions ~(t) - Xt) = (y(t), Z(t)), t t < tt2 k(t) - x(t) - (y'(t), zk(t)) - f(t,yk(t), uk(t)), tlk t < t2k k=1, 2,o o (17)

34 are defined almost everywhere and are L-integrable. We have to prove that [t, y(t), Z(t)] e A for every t1 < t < t2, and that there is a measurable control function u(t), t1 < t < t2, such that W(t) - X'(t) (y'(t), Z'(t)) = f(t, y(t), u(t)), (18) u(t) E U(t,y(t)), for almost all t e [t1, t2]. First, P(yk y)-O as k - O hence tlk tl t2k t2 If t E (tl, t2), or tl < t < t2 then tk < t < t2k for all k sufficiently large, and (t, Yk(t) e Ao. Since Yk(t) -y(t) as k - oo and A is closed, we conclude that (t, y(t)) e A for every tl < t < t2 and finally (t, y(t), Z(t)) e A x En or (t, (t)) A, t t For almost all t e [t1, t2] the derivative X'(t) = [y'(t), Z'(t)] exists and is finite, S'(t) exists and S'(t) = 0, and zk(t) - z(t). Let to be such a point with t1 < t < t2 Then there is aa > with t1 < to - <to + a < t2 and, for some k and all k > ko, also tlk < t < t + a < t2k. Letx X(to) (Yo Z), or = y(to), Z0 = Z(to) Let z = z(to), So S(to) We have S'(to) = 0, hence z'(to) exists and z'(to) = Z'(to)o Also, we have zk(t) - z(t o) We have Yk(t) - y(t) uniformly in [to - a, to + a], and all functions y(t), Yk(t) are continuous in the same interval. Thus, they are equicontinuous in [t - a, t + a] Given e > 0, there is a 6 > 0 such that

35 t, t E [to - o, t ], l t-t't < 6,k k k, implies ly(t) - y(t') i /2, Yk(t) - Yk(t)l < E/2. We can assume 0 < 6 < a, 6 <oE For any h, 0 < h < 6, let us consider the averages h mh h f (t0 + s)ds - h1[X(t + h) - X(to))] 0 (19) h mhk h1 kt^ + s)ds h [kto + h) - xk(to)) where X (y, Z), k = (yk' zk) Givenr > O arbitrary, we can fix h, O < h < 6 < a, so small that Imh- (t)i < K, S(t + h) - S(t) l < h/4 0 h This is possible since h J i (to + s)ds - f(t ) and 0 [S(t + h) - S(t )] h 0 as h 0. Also, we can choose h, in such a way that Zk(to + h) z(to + h) as k- + oo This is possible 0 since zk(t) " z(t) for almost all t1 < t < t2o Having so fixed h, let us take kI > k so large that Yk(to) - y(t ) tk (t + h) - y(t + h)t < min[r h/4, e/2], I zkt.) - z.(t) I, i zktO + h) - z(to +h) i ( r h/8.

36 This is possible since Yk(t) - y(t), zk(t) - z(t) both at t = t and t = t +h. Then we have h- [yk(to + h) - yk(to)] - h[y(t + h) -(t) < Ih [yk(to+ h) - y(t + h)]l + Ih [yk(to)- Y(to)]I < h- ( h/4)+ h- 1( h/4) 1/2. Analogously, since z = Z + S, we have Ih- zk(to h) k(to)- h (t+ h) - Z(to)]l h zk(to+ h) - zk(to) - h [z(to + h) - z(to)] +h (t+) -[S(th) - S(t] 0 O < Ih- [zk(to +h) - z(to +h)] + Ih [zk(to)_ Z(to)] + {h- 1[S(to + h) - S(to)] < h (77 h/8) + h- (7 h/8) + h- (7 h/4) = 7/2. Finally, we have Imhk -mhl Ih [xk(t +h) - xk(to)] - h [X(to + h) - X(to)] < Ih-'yk(t + h) - yk(t)] - h[y(t + h)- y(t)] +

37 + Ih l[zk(t + - Z h[Z(t + h) - zZ(to)] < /2+ +/2 =/ We conclude that for the chosen value of h, < h < 6 < r, and every k > k we have mh - to! < ]mhk - mh <m,_ jYk(t) -y(t0)l < E/2. (20) For 0 < s <h we have now Yk(to + s) - y(t) I < l Ykit + s) - Yk(to) l + Yk(to) -Y(to) < e/2 + /2 E, to s) - to < h < 6 < E f(to + s, YkVto (+ ), ukt + S)) E Q(to + S Yk(to + S)) Hence, by definition of Q(to9 yo 2e), also Ok(to = s) - f(t s y s, ukk(t + s) ) e Q(tuk yo, 2e). The second integral relation (19) shows that we have also mhk e cl co Q(tO yog 2e), since the latter is a closed convex seto Finally, by relation (20), we deduce

38 I(to) mhk < l(t - mhl + Imh mhk < 2i, and hence 0(to) e [cl co Q(to0 yo, 2e )]27o Here 1 > 0 is an arbitrary number, and the set in brackets is closed. Hence )(to) e cl co Q(to0 Y 2e ), and this relation holds for every e > 0. By property (Q) we have.(to)e e clcoQ(to, y, 2e) =Q(to, ), where yo = y(to), and Q(to, yo) = f(to y, U(t, y))O This relation implies that there are points u - u(to) e U(to, yo) such that 0(to) = f(to, y(to) u(to))o This holds for almost all to e [t1 t2]o Hence, there is at least one function u(t), defined a. eo in [t1, t2] for which relation (18) holds a. e. in [t1, t2]. We have to prove that there is at least one such function which is measurable. The proof is exactly as the one for Closure Theorem I, where we write y, Yk instead of x, xk, and will not be repeated here. Closure Theorem II is thereby proved. REMARK: The proof of Closure heorem I is a modification due to Cesari [4a] of the proof of an analogous statement by A. F. Filippov [8] for compact instead of closed sets. Both the statement and proof of Closure Theorem II bearing on singular functions

39 are due to Cesari [4a]. 8. Notations for the optimal control problem, We shall again use the notations of No. 1. It will be convenient to write the problem in a slightly different form. First we introduce the auxiliary variable x satisfying the differential equation and initital value dx~/dt = f(tx(t) u(t)), x~(tl) - 0 x~(t) AC in [t1, t2 Then t2 x(t2) = f f(t, x(t), u(t))dt =I[x, u]. (2 1) t If we now denote by x the (n+l)-vector x = (x, o, x), and by f (t, x, u) the (n+l)-vector function f (t, x, u) (fo0 fl "' fn ) then the problem of minimum discussed in No. 1 reduces to the determination of a pair [x(t), u(t)], t1 <t <t2, satisfying the differential system dx/dt - f(t,x(t), u(t)) a. eo in [t1, t2], (22) the boundary conditions (t, x(tl), t2, (t2)) e B, x(t) = 0, (23) and the constraints (t, x(t))~ eA, u(t) e U(t, x(t)), t E[tl, t2]

40 for which x~(t2) has its minimum value. Here x(t) = (x, o. x x(t) = (x,x, x ), and the present formulation corresponds to a transformation of the Lagrange type problem of No. 1 into a problem of the Mayer type. We shall now consider for each (t, x) e A, the sets Q(t, x), Q(t,x), 4(t, x) defined as follows: Q(tx) = f(t,x,U(t,x)) = [zlz=f(t,x,u), uE U(t,x)] c En, Q(t,x) - f(t,xU(t,x )) - [z (z, z)lz =(t,x,u), u U(t,x)] [z = (z, z) z =fo(t, x, u), z=f(t, x, u), u U(t, x)] C Eni Q(t,x) [z = (z, z) z > fo(t, x, u), z - f(t, x, u), u e U(t, x)] c Eni The main hypothesis of the existence theorems which we sha_ state and prove below is that the set Q (t, x) is convex for each (t, x) e Ao For free problems the sets Q, Q, ( - thought of as subsets of the z u-space are Q(t,x) = E Q(t x) = [z (z, u) z = f (t, x, u), ue En c E 1' Q(t,x) = [z (z, u) z > f (t,x,u), ue En] c E' -'

41 Thus, the convexity of Q reduces to the usual convexity condition of fo(t x, u) as a function of u in E — a condition which is familiar in the calculus of variation for free problems. The proof of this equivalence is to be found in [7]. We mention here that a function p(u), u e En, is said to be convex in u, provided u, v E En, 0 a < 1, implies 4(au + (1 - a)v) < a ~(u) + (1 - a) q(v)o 9. Statement of existence theorem I Existence Theorem I (Cesari [4a]) Let A be any compact subset of the tx-space E1 x En, and for every (t, x) e A let U(t, x) be a closed subset of the u-space Em. Let M be the set of all (t, x, u) with (t, x) e A, u e U(t, x), and let f(t X, u) (f f (fo fg, f) be a continuous vector function on M. Assume that for every (t, x) e A the set Q(tx) - {z - (z9 z) lz > fo(t, x,u), z - f(t x, u), u e U(t,x)} C E n+1 is convex. Assume that U(t, x) satisfies property (U) in A, and that Q (t, x) satisfies property (Q) in A. Assume that there is a continuous scalar function (0, 0< < < + oo, with D(O/C - + oo as C- + oo, such that fo(t, x, u) > b(1 ul) for all (t, x, u) E M, and that there are constants C, D > 0 such that

42 f(t, x, u) l < C + D l u for all (t, x, u) e M. Then the cost functn tional lx, u] f t fo(t, x, u)dt has an absolute minimum in any 1 nonempty complete class n of admissible pairs x(t), u(t). If A is not compact, but closed and contained in a slab [to < t < T, x E En], to, T finite, then Existence Theorem I still holds under the additional hypotheses that (a) f - f f +.0 +x < N[(x) +0 +(xn +1] for all (t, x, u) e M and some constant N > 0, and (b) each trajectory x(t) of the class O contains at least one point (t*, x(t*)) on a given compact subset P of A (for instance, the initial point (t*, x(t*)) is fixed, or the end point is fixed). If A is not compact, nor contained in a slab as. above, but A is closed, then Theorem I still holds if the hypotheses (a), (b)andf (c) are satisfied. (c) fo(t,x,u) > t > 0 for all (t,x,u) e M with itl > R and some constants A > 0, R > 00 Finally, condition (a) can be replaced in any case by condition (d)o (d) fo(t,x,u) > E f(t, x, u) for all (t,x, u) e M with xi > F and for some constants E > 0 and F > 0. Furthermore, when A is compact but closed, the conditions f > (Ilul), Ifj, < C + D luI above can be replaced by the

43 following condition (g): (g) for every compact subset Ao of A there are functions b as above and constants C > 0, D > 0 (all may depend on A ) such that fo > 4o_(lul),.j f < Co + D lul for all (t, x, u) E M with (t, x) A 0 PROOF: We have'(I > - M for some number M > 0, hence 4( + M > 0 for all C > 0, and fo(t, x, u) +M > 0 for all (t, x, u) e M'. Let D be the diameter of A. Then for every pair x(t), u(t), t < t < t2, of S we have t2 t I[x,u] = 2 f0 dt > S2 (ul)dt > -DM > -oo (25) t1 t 1 1 Let i = inf lx, u], where inf is taken over all pairs (x, u) E Q. Then i if finite. Let xk(t), ukt), tlk < t < t2k, be a sequence of admissible pairs, all in rS2 such that I[xk, uk] i as k - + oo. We may assume t2k i < l[xku] - fk fo(txk(t), uk(t))dt < i+ k tlk 0 be any given number, and let oa 2 e (DM + il + 1). 0 Do+li )1

44 Let N > 0 be a number such that 4C)/C > a for > N. Let E be any measurable subset of [tlk, t2k] with meas E < r7 -1 -1 e 2 N Let E1 be the subset of all t e E where uk(t) is finite and uk(t) < N, and let 2 =E - E1. Then iuk(t) < N in E and (l uk(t)l) luk(t)l- >i or lu < C(lukl), e. in.E2. Hence (E) luk(t) dt = (E1+E2) lk(t)l dt < N meas E1 + a(E2) i uk(t) )dt < N meas E + a(E2) f [Iuk(t)l) + Mo]dt (26) < N7 + a S k [(luk(t)l) + Mo]dt tlk as 4 + M > O for 0 < < + oo. Asf > b(IluJ) for each (t, x, u) e M, one has that t2k (E) S luk(t) dt < N + a f 2ka [fo(t, xk(t), uk(t)) + Mo]dt lk < N7 + a(DM + il + 1) < e/2 +e/2 = e. This proves that the vector functions uk(t), t1k < t < t2k, k=1, 2,.. are equiabsolutely integrable. From here we deduce

45 (E) lx'k(t) dt (E) S f(txk(t), uk(t))ldt (E) [C + Dluk(t) dt C meas E +D (E) f luk(t) dt as f(t, x, u) < C + D ul for each (t, x, u) e M and this inequality and the equiabsolute integrability of the vector functions uk(t), t1 < t < t2, prove the equiabsolute continuity of the vector functions xk(t) t1 < t < t20 Now let us consider the sequence of AC scalar functions xk(t) defined by t Xk) o (, Xk(r) uk(r))d t1 < t <2k (27) tlk Then xk(lk) 0, X(t 2k) [Xk, Uk i as k -co and i < xk(t2k) < 1 +k < ii+1 k=l 2, o If vkt) f f(t), uk(t)), t1k < t < t2k~ then we define the vt) - fo(t xk(t)' uk(t)b, -k 2kk functions vk(t), vk(t) as followsvk(t) = -Mo0 vkt) - v+M f(t xk(t), uk(t))+M o > 0 Then vk(t) < 0, v+(t) > 0 ao e. i [tlk tkE and we define

46 t t yk(t)- vk(t)dt, Yk(t) f vke(t)dt tk tlk tlk < t < t2k k=12 eoo Since vk(t) - M, we have yk(t) = - M (t-tlk) < 0, and the functions yk(t) are monotone nonincreasing and uniformly Lipschitzian with constant M. On the other hand, the functions yk(t) are nonnegative, monotone nondecreasing, and uniformly bounded since 0 Yk (t) = (yk(t2k) - Y(t k)) - Yk t2k -(t < i + 1 + M o(t2k tlk) < DMo + il + 1. By Ascoli's Theorem we first extract a sequence for which (xk(t), Yk(t)), tlk <_ t K t2k converges in the p-metric toward a continuous vector function (x(t), Y (t)), t1 < t < t2o Here x(t) is AC because of the equiabsolute continuity of the vector functions x(t), tlk < t t2k and Y (t) - -Mot-tl, Y (t1) 0. Then we apply Helly's Theorem to the sequence yk(t), t < t t t k=1, 2, oo, and we perform a successive extraction so that the corresponding sequence of yk(t) converges for every t1 < t < t toward a function Y(t), tl < t < t2 which is rnoregative, monotone nondecreasing, but necessarily continuous. We define Y (t) at t= by taking Y0(t) - 0, and at t2 by continuity at t2, o i i^"~

47 because of its monotoneityo Thus 0 < Yo(t) < DMo + il + 1, tl< t < t2 Finally, Y (t) admits of a unique decomposition Y (t) 0 Y(t) + Z(t), tl < t < t with Y"(t1) 0, where both Y+(t), Z(t) are nonnegative monotone nondecreasing, where Y+(t) is AC and Z?(t) - 0 a. e. in [t1, t2]. If Y(t) - Y-(t) + Y+(t), we see that yk(t), tlk < t t k=1 2,.., converges for all t1 t < t2, toward x (t) - Y(t) + Z(t), where Y(t) is a (scalar) AC function, - DMo < Y(t) < DMo + il + 1, Y(t) - 0 Let us prove that Y(t2) < i. For the subsequence [k] we have extracted last, we have t2k - t, Xkt2k) -i, xkt2 Yk (t2k) + Ykt2k If t2 is any point tl < t < t2, t2 as close as we wish to t2, then t2 < t2k for all k sufficiently large (of the extracted sequence), since t2k - t2 We can assume k so large that t2 < t2k 2 t2 t2k < 2 It2 - Then Ykt2- Ykt2k) - Mo2it - 2 t2k 2Mo 2 t2 Since Yk(t) is nondecreasing, we have y(t2) < Y(t2k, and finally

48 2) + Yk(t 2) < Y(f2) + Yk(tk) Yk(t2k) + Yk(tk)+ Ykt2) - Yk(t2k) i < Yk(t2k) +2Mo t2 - t21 0 0 — 1 where Y(t2k) i as k- + oo and y(t2) < i +k Hence k 2k k 2kk(t ) +Yk(t2) 0, Thus Y(t2) = Y-(t2)+ ) < i +2 Molt2 - t21. As t2 -t2 - 0, we obtain Y(t2) < i, since Y is continuous at t2. We shall apply below Closure Theorem II to an auxiliary problem, which we shall now define. Let u = (u, u) = (u, u,.., u) let U(t, x) be the set of all u e Em with u (ul, o o, um) e U(t, x), 0 0 01 n u > f (t, x, u), let x x) (x, ) (x, x l, x ), let f (t, x, u) (f, f) - (f, f, o ) fn) with f u Thus, depends only o (t, x, u (instead of (t, x, u), and U depends only on t, x (instead of (t, x)o Finally, we consider the differential system dx/dt = f(t,x,u) or dxo/dt u (t), dxi/dt f.(t, x, u), i=l, 2, o o o, n,

49 with the constraints u(t) e U(t, x(t)), or u (t) > f(t, x(t), u(t)), u(t) e U(t, x(t)), ~- 0O a. eo in [t1, t2], with moreover x (t) 0, (t, x(t)) e A, and (x, u) e E2o We have here the situation discussed in Closure Theorem II, wherexreplaces x, x replaces Y, x replaces Z, n+1 replaces n, n replaces s, hence (n+1) - n = 1 replaces n-s. For the new auxiliary problem the cost functional is t t Note that the set Q (t, x) = f (t, x, U(t, x)) of the new problem is the set of all z - (z, z) e En+ such that z - u, since f o u z = f(t, x, u), u > f (t, x, u), u e U(t, x). Thus, the sets U, ( for this auxiliary problem are the sets U,Q considered before. We consider now the sequence of trajectories xk(t) = [xk(t) xk(t)] tlk < t < t2k, for the problem J[x, u] corresponding to the control function uk(t) = [uk(t), uk(t)], with u(t) = fo(t, xk(t), uk(t)), uk(t) E U(t, xk(t)), and hence uk(t) e U(t, xk(t)), t1k < t < t2k k=1, 2, o.o The sequence [xk(t)] converges in the metric p toward the AC vector function x(t), while x?(t) - x~(t) as k - + oo for all t E (tl, t2), and

50 x~(t) = Y(t) + Z(t), where Y(t) is AC in [t1, t2] and Z'(t) = 0 a. e. in [t, t2]o By Closure Theorem II we conclude that X(t) = [Y(t), x(t)] is a trajectory for the problemo In other words, there is a control function u(t), t1 < t < t2, u(t) = (u (t), u(t)), with dY/dt = u~(t) > f (t, x(t), u(t)), u(t) U(t, x(t)), (28) dx/dt f(t, x(t), u(t)), ao e in[t, t2], and t2 0 i > Y(t2) = J[x,u] = u (t)dt. (29) t1 First of all [x(t), u(t)] is admissible for the original problem and hence belongs to f since by hypothesis Q is complete. From this remark, and relations (28) and (29) we deduce t2 tz i < [xu] - S fo(t, ) x u(t))dt << io ti t Hence all < signs can be replaced by = signs, u~(t) - f(t, x(t), u(t)) a. e. in [t1, t2], and I[x, u] = i This proves that i is attained in f Thus, Existence Theorem I is proved in the case that A is compacto

51 Let us assume now that A is not compact but closed, that A is contained in a slab [t < t < T, x e E ], to, T, finite, and that conditions (a) and (b) holdo If Z(t) denotes the scalar function Z(t) = x(t) 1 + 1, then the condition xo f < N( x2 + 1) implies Z' < 2NZ and hence, by integration from t* to t, also 1 < Z(t) < Z(t*) exp {2Nlt* - tl}o Since [t*, x(t*)] e P where P is a compact subset of A, there is a constant No such that Ixl < No for every x P, hence 1 < Z(t*) < N+1, and < Z(t) < ( + 1) exp {2NT - t)} Thus, for t < t < T, Z(t) remains bounded, and hence Ix(t) < D for some constant Do We can now restrict ourselves to the consideration of the compact part Ao of all points (t, x) E A with t < t T Ixl < Do Thus, Theorem I is proved for A closed and contained in a slab as above, and under the hypothesis (a) and (b). Let us assume that A is not compact, nor contained in a slab as above, but closed, and that hypothesis (a), (b) and (c) hold. First, let us take an arbitrary element (x(t), u(t)) e Qand let j - I[x, u]o Then we consider a bounded interval [a, b] of the t-axis containing the entire projection Po of P on the t-axis, as well as the interval [-R, +R.] In the slab [a < t < b, x E En] conditions (a) and (b) hold and hence by the previous argument there is a D (constant) such that { x(t) I < D and we can confine our attention from

52 A n [a < t < b, x E E ] to some compact part of this set, say Ao For A there is a function 4(D, 0 < < + o, and constants C, D > 0 such that f (t, x, u) > ((I u) and i f < C + D oul for each (t, x, u) e M with (t, x) e Ao, Therefore by previous reasoning, there is an Mo (this argument still holds even if %, CO, D and M depend on Ao) such that f (t, x, u) > Q4 ul ) > - M and fl < CO + Do u for each (t, x, u) E M with (t, x) e Ao Condition (c) guarantees that for each (t, x, u) e M with (t, x) f A n [a < t < b, x E E ] and hence for each (t, x, u) e M with (t, x) A, there is a i > 0 such that fot, x, u) >' > Oe Now let = I [ljl + 1 + (b-a)Mo], let [a', b'] denote the interval [a-&, b+&]. Then for any admissible pair (if any) (x(t), u(t)), tl < t < t2, of the class D, whose interval [tl, t2] is not contained in [a', b], there is at least one point t* e [t1, t2] with (t*, x(t*)) e P, a < t* < b, and a point t e [t1, t2] outside [a', b']o Hence [t1, t2] contains at least one subinterval, say E, outside [a, b], of measure > f. Then l[x, u] > &g - (b-a)M ~ I j + 1 > i + lo Obviously, we may disregard any pairs x(t), u(t), t < t < t2, whose interval [tl, t2] is not contained in [a', b']o In other words we can limit ouselves to the closed part A' of all (t, x) c A with a < t< b'o We are in the previous situation, and Theorem I is proved for any closed set A under the hypothesis (a), (b) and (c)o Finally, we have to show that condition (a) can be replaced by condition (d) in any caseo It is enough

53 to prove Theorem I under the hypothesis that A is closed and contained in a slab t < t < T, t, T finite as above, and hypothesis (b) and (d). First let us take F so large that the projection P* of P on the x-space is completely in the interior of the solid sphere Ixi < F, and also so large that F > T - to Let x(t), u(t), t < t < t2 be any arbitrary pair contained in f and let j denote the corresponding value of the cost functional. Let L E 1FM + ljl + 1], and let us take F F + Lo If any admissible pair x(t), u(t), t1 < t < t2 of 2possesses a point (to x(t )) with Ix(to)l > Fo, then x(t) possesses also a point (t*, x(t*)) E P with I x(t+) < Fo Thus, there is at least a subarc F: x x(t), t' < t < t" of x(t) along which I x(t) > F and x(t) passes from the value F to the value F = F +Lo Such an arc r has a length > Lo If G t 11 t2][t', t"], then for t E G, Ixl > F, (t, x) e A, f (t, x, u) > 0 and letting A be the compact part of A with t E G, Ixl < F we have, as before, an M > 0 such that f + M > O for each (t, x u) E M, (t, x) c A o Thus, one has f > - M for each (t, x, u) E M with t E Go Then

54 t t" I[x, u] 2 f dt (G) f dt + f dt t t't ti >- (T - to)M + E fldt tt - FM + E j ifldt > -FM +EL t'0 > ljl +1 > i+l. As before we can restrict ourselves to the compact part A' of all points (t, x) e A with t < t < T, Ix < F. The case where A is closed, A is not contained in any slab as above, but conditions (b), (c) and (d) hold can be treated as before. The case where A is not compact and condition (g) holds, also can be treated as before. Theorem I is thereby completely provedo 10o A few corollaries. Corollary 1 (Ao F. Filippov's existence theorem for Pontryagin's problems). As in Theorem I, if A = E1 x EnW f (t, x, u) - (fo0 f) A (fo0 f19 o o fn) is continuous on M, U(t, x) is compact for every (t, x) in A, U(t, x) is an upper semicontinuous function of (t, x) in A, Q(t, x) = f(t, x, U(t, x)) is a convex subset of E +1 for every (t, x) in A, conditions (a) and (c) are satisfied, and the class D of all admissible pairs for which x(t1) = x1, x(t2) =- x2

55 t1, x1, x2 fixed, t2 undetermined, is not empty, then I[x, u] has an absolute minimum in Po PROOF, This statement is a corollary of Theorem Io Indeed, under hypothesis (c) we can restrict A to the closed part A of all (t, x) e A with a' < t < b', and I xl < N for some large N. If M is the part of all (t, x, u) of M with (t, x) E A, then the hypothesis that U(t, x) is compact and an upper semicontinuous function of (t, x) in Ao certainly implies that U(t, x) satisfies property (U) in A and that Mo is compact (No. 5, (ii) and (iii))o Also, since Q(t, x) is convex for every (t, x) by hypothesis, we deduce that Q(t, x) is an upper semicontinuous function of (t, x) and satisfies property (Q) (No 5, (v) and (vi))o Also, Q (t, x) is closed, convex, and satisfies property (Q) by force of Lemmas (viii) and (x) of Noo 4. Finally, since Mo is compact, the growth condition f > b and the remaining condition fl < C + Dl ul are trivially satisfied. Thus, all conditions of Theorem I are satisfied, and Filippov's theorem is proved to be a particular case of Theorem L Corollary 2 (the Nagumo-Tonelli existence theorem for free problems)o If A is a compact subset of the tx-space E1 x E, f (t, x, u) is a continuous scalar function on the set M A x E, for every (t, x) e A, f (t, x, u) is convex as a function of u in A and there is a continuous scalar function <(I), 0 < C + c, with ()/C - +oo as C -+ oo such that fo(t, x,u) > lul) for all

56 (t, x, u) E M, then the cost functional t2 I[x] - f (t, x(t), xI(t))dt t has an absolute minimum in any nonempty complete class Q of absolutely continuous vector functions x(t), t1 < t < t2, for which fo(t, x(t), x'(t)) is L-integrable in [t1, t2]o If A is not compact, but closed and contined in a slab [t < t < T, x e E ], to, T finite, then the statement still holds under the additional hypotheses (rl)fo> C ul for all (t, x, u) e M with x > D and convenient constants C > 0, D > 0; (T2) every trajectory x(t) of I possesses at least one point (t*, x(t*)) on a given compact subset P of A. If A is not compact, nor contained in a slab as above, but A is closed, then the statement still holds under the additional hypotheses (T1), (r2), and (T3) fo(t, x, u) > t > 0 for all (t, x, u) e M with I t > R, and convenient constants g > 0 and R > 0. PROOF: The free problem under consideration can be written as an optimal control problem with m = n, fi - u., i=1l,., n, U(t, x) = Em En, so that the differential system reduces to dx/dt - u, and the cost functional is

57 t2 t2 I[x, u] S fo(t, x(t), u(t))dt - fo(t, x(t), x'(t))dt ti t First assume A to be compact. Then the set ( (t, x) reduces here to the set of all z = (z z) e E, with z > f (tx z), z E n+1 wi th >fo n where fo is convex in z, and satisfies the growth condition f0 > (lul) with 4)/C-+oo as ~-+ oo By lemma (x) of Noo 5, fo is normally convex in u, hence quasi-normally convex, and by lemma (viii), part (j) of No. 5, Q satisfies property (Q) in A. Thus, all hypotheses of Theorem I of Noo 9 are satisfied. If A is closed but contained in a slab as above then condition (a) of Theorem I reduces to u. x < C(l x + 1) which cannot be satisfied since we have no bound on Uo On the other hand, the condition (d) f > El f for some E > 0 reduces here to requirement (T1) and condition (b) to requirement (T2)o Finally, if A is not compact, nor contained in a slab as above, but A is closed, then requirement (c) of Theorem I reduces to requirement (73)0 All conditions of Theorem I are satisfied, and the cost functional I[x, u] I[x, x'] has an absolute minimum in o2.

Chapter II Further Existence Theorems 11. An existence theorem with uniform growth of f o Let us assume for the moment that A is compact. The condition of Theorem I (a) fo(t, x, u) > ( lul) for each (t, x, u) e M where A(), 0 < < + oo, is a continuous scalar function satisfying (C)/C - +oo as - +oo; If(t,x,u) < C +DIul for all (t, x, u) e M and some constants C, D > 0; is usually called a "growth condition." This condition (a) obviously implies (j) fo(t,x,u) lul- +oo as ul -+oo, ue U(t,x) for each (t,x) e A; If(t,x,u) < C + Diul for all (t,x,u) e M and some constants C, D > 0. This condition (P) is also a "growth condition", and examples show that-in general —(3) is a weaker condition than (a). An example of this is given below. Nevertheless, there are situations where (a) and (p) (for A compact) are equivalent. One of these situations concerns free problems, that is, m=n, f=u, U=E n For these problems, with fo convex in u for every (t, x) in A, conditions (a) and (i3) are equivalent (Tonelli [23a] for f of class C; LTurner [24] 58

59 for fo only continuous as assumed here)o This case of equivalence, together with another relevant case of equivalence will be obtained below as a consequence of a number of lemmaso Example: Let m-n 1, A [(t,x) < t <, 0 < x <], U(t, x) E1 for each (t, x) E A, f(t, x, u) = u and fo(t,x,u) -tu for 0 u<21t(1-t) 0< t < -12-1 -1 t[u-t(l-t) ] for u > 2 t(1-t) 0 < t< 2 u for < u<+ oo, t=1 fo(t, x,u) f (t, x, -u) for - oo < u < + ot, <O t l< 0 ~ fo(t x u) 0 < t < i' / x'', / t 1 t t t - l(T-t) -2 I -2 (T —T 1-t Figure 1 An example where f% ul- -+ o as l ul -+oo, u E pointwise in A, but not uniformlyo

60 Hence fo t,x o) - f(tx,t(l-t) ) 0 fo(t,x, 2 t(l-t)1) - 2 2t3(1-t)2 for 0 < t < 1, 0 <x<, and f is a continuous function of (t, x, u) in A x E. The second part of both conditions (a) and (3) is satisfied by C - 0, D = 1 2 as Ifl - lul in A x E For t =, we have f = u and for O < t < 1, we have f = (u - t(l-t)- )2 for ul sufficiently large. Hence f /Iui + oo as ul -+ oo, for every (t, x) e A, and condition (3) is satisfied. On the other hand, for every lul, there are (t, x) E A with f 0, namely all (t,x) with lul t(1-t) 1, or t lul(lul + 1) - Hence, a relation f (t,x,u) > (lul) can be satisfied for all (t, x, u) e A x E1, only with 14 < 0, and condition (a) is not satisfied. In this example fo is not convex in u, and U(t, x) E1. An analogous example with fo convex in u can be obtained by taking A, f as above, by taking U(t, x) [ul t(1-t)- < u < + oo] for 0 < t < 1, U(1, x) [u 0 u< + oo] and by defining f as above. Then, for every (t, x) c A, fo is convex in u for u E U(t, x), condition (3) is satisfied and condition (a) is noto A condition slightly stronger than (1) has been taken into consideration, say (y), and we state it here again for A compact.

61 (0) fo(t,x,u) iul 1-+oo uniformly for (t, x) A as Iul - + oo, u e U(t, x); f(t, x, u) I < C + D u for each (t, x, u) e M and some constants C, D> 0. Obviously, condition (a) implies condition (y). The following three lemmas show that (a) is equivalent to (y) when A is compact. Lemma 1: Let A be a fixed compact subset of the tx-space E1 x En, and for every (t, x) e A let U(t, x) be any subset of the u-space Emo Let fo(t, x, u) be a continuous scalar function on the set M of all (t, x, u) with (t, x) e A, u e U(t, x), and let Z be the subset of all nonnegative z with z = I ul for some (t, x, u) e Mo Then ul lfo(t, x, u) - + o as lul -+ co, u E U(t,x), uniformly onA, if and only if there is a scalar function 4(z), z e Z, bounded below, with 4(z)l z - 4 o as z - + oo, z Z, and f (t,x, u) > ( ul) for each (t, x, u) e M. PROOF: Suppose such a 4(z) exists with the above properties. Thenforu 0, f lui > 4(lul)lul-1 andhence f ul u - + c as lul - + oo, ue U(t,x), uniformly on A Suppose fol ul - +oo as I ul - +, u e U(t, x) uniformly on Ao For each 0 fixed z e Z, let (z) = inf fo(t, x, u) for all (t, x) e A, ul - z, u e U(t, x). If 4z)/z does not approach +oo as z - +, z e Z, then there exists an N > 0 and a sequence (t, x, u ) E M such that fo(t, x^, ) u - 1I < N, lu - +oo as v-+oo OL) 1) V V V

62 As f lul _+ oo as lu - +oo, ue U(t,x), uniformly onA, then for N* - N + 1 there exists a constant A > 0 such that f i ul-1 > N* = N + 1 for all u e U(t, x), I ul > A and this last inequality holds for any (t, x) e A. Therefore, N + 1 = N* < f (tv, x, u ) lu -1 < N for v large enough so that I u i > A. This is a contradiction and thus, (z)/z - + oo as z - +oo, ze Z. Clearly fo(t, x u) > 4(1ul) for each (t,x,u) E M by the definition of 4z), z e Z. Since 4(z)/z - co as z - + oo, z e Z, we also conclude that (z) - +oo as z + oo, z e Z and hence there is some zo such that b(z) > 0 for all z > z, z e Z. Now the set S' of all points (t, x, u) e M where u belongs to a sphere S = [ul ul < z ] is a compact set, as the intersection of the closed set M with the compact set A x S. Thus, f is continuous on S', hence bounded there, say fo! < Mo, and thus ((z) > -M for each z e Z n [o, z]. This proves that.(z) is bounded below in Z. Lemma 1 is provern Lemma 2: q (z) is a scalar function of z on a subset Z of [o, + oo), if 41(Z) is bounded below, and if (1(z)/z -+ o as z - + oo, z e Z, there is also a scalar function <4z), continuous on [o, + oo) such that 1 (z) > 4(z) for each z e Z and I(z)/z - + oo as z -+ 0.

63 PROOF: Let A be a bound below for l(z), z E Z. Now for each n=1, 2,... there exists a A(n) > 0 such that 4 (z)l z > n when z > A(n), z E Z and we can assume that A(n) < A(n+l), n=l, 2, o.. Let T2(z), - 1 < z < + oo, be the function defined by D2(z) 1 A0 when z [-1, A(1)) 42(z) = nz when z e [A(n), A(n+l)) for n=l, 2, o Clearly 4l(z) > 42(z) for z e Z and D2(z)/z - + oo as z -+ oo, and <2(z) is monotone nondecreasing in [-1, + oo). For any compact subset K of [-1, + o), 42(z) is Lebesque integrableo Let 4(z), 0 < z < + oo, be the function defined by z (z) f 42(zs) dz?, 0 K z < +oo z-1 As [z-1, z] is a compact subset of [-1, + oo) for each z e [0, + oo), 4z) exists for each z E [0, + oo). On the other hand, -1 - 1 1 Z)/ z 1 z" 4D2(z)dz' > (z-1)z (z-l) 2(z-l) z-l > - n for all z > 2 withz e [A(n) + 1, A(n+l) + 1) Thus z)/z- + oo as z- + oo Finally, by the monotoneity of "2 we have

64 z 4z) - (f 2(z')dz' < 2(z) < 1(z) z-1 for all z c Zo Since D2(z) is L-integrable in any finite interval of [-1, + co), we conclude that 4(z) is a continuous function in [0, + oo). Lemma 3: Let A be a fixed compact subset of the tx-space E1 x Eno If fo(t, x, u) is a continuous scalar function on M, the set of all (t, x, u) with (t, x) e A, u c U(t, x), then f ul1 + oo as I ul - + co, u c U(t, x), uniformly on A, if and only if there exists a continuous scalar function 4(z) on [0, + oo) such that 4(z)/z - + oo as z - + o and fo(t, x, u) > El ul) for each (t, x, u) E M. PROOF: The sufficiency is obvious. The necessity is an immediage consequence of lemmas 1 and 2. Having proved that properties (a) and (y) are equivalent for the general optimal control problem formulated as a Lagrange problem, we proceed now to the consideration of two special cases, where it is possible to replace property (a) by the weaker property ((3) in the statement of Existence Theorem II. These two cases will be the object of lemmas 4 and 5 below0 In both lemmas 4 and 5 we shall assume that the control space is a fixed closed subset of the u-space Em, and therefore that M has the form M A X U where A is a given compact subset

65 of the tx-space E1 x E o Let f (t, x, u), (t, x, u) A U be a given scalar function. We shall say that f (t, x, u), (t, x, u) A X U is a uniformly continuous function of (t, x) in A with respect to u, if for each E > O and (to x) e A there is a 6 6 (to xo, ) > 0 such that i fot,, u) - f0(to0 x0, u) < e for all (t, x) e A, u c U, (t, x) e N6 (t0, x0 ) (30) Lemma 4 Let A be a compact subset of the tx-space E1 X En, let U be a fixed closed subset of the u-space Em, and let fo(t, x, u), (t, x9 u) e A x U, be a continuous scalar function, continuous on M and a uniformly continuous function of (t, x) in A with respect to uo Then f (t, X u)/iul + oo as ul -+ oo, u U, for every (t,x) E A, if and only if f (t,x, u)/I ui + oo as lul + oo, u e u, uniformly on A PROOF: The sufficiency is obviouso Suppose fo/l ut does not tend to + oo as ul -+ oo u u U, uniformly on Ao Then there is a sequence (t tv, x u ) E M Ax Uwith lu i -+oo as v-+oo andf (t,x, u)lu l < N v 0 v Vfor some constant N. The sequence t,x ), 1 29 o o o has a convergent subsequence, as A is compact. Let (t, x ) E A be the limit of this convergent subsequence. As fo(t, x, u) is a continuous function on M and a uniformly continLuous functiorn of (t, x) m A with respect to u, for (E 1, there

66 is a 6 = 6 (to, x,1) > 0 such that Ifo(t,x,u) - fo(t, xou) < 1 when (t, x, u) c M with (t, x) c N6 (to, x )o Given any constant L > 1, then lul > L implies that lul > L > 1 and Ifo(t,x,u)j ul- fo(to, Xo u) lu-ll < u luI < L1 <1 (31) when (t, x, u) e M with (t, x) e N6 (to, x ). Since (t, x ) - (to, x) as v + oo then (t, Xv) e N6 (to, x) for all vsufficiently large, and f (t x, u )/luvI < N together with (31) implies that fo(to0 x,u ){uvI -1 < N+1 for all v sufficiently large, and as u I - + oo as v - + oo, u E U, this contradicts the hypothesis fo(t, x, u)lul- - + oo as lul - + o, u e Uo Lemma 4 is proven. We have already shown by one of the examples in No. 11 that there are functions fo(t, x, u), (t, x, u) e A x U, for which fo(t, x, u) lul _ + oo as lul - +oo, u e U, for every (t, x) e A (compact), but for which the same limit does not occur uniformly in Ao Obviously, in the example given inNo. 11 f0 is continuous on M, but is not a uniformly continuous function of (t, x) in A with respect to u. On the other hand, lemma 4 concerns a class of functions which is not empty. Below we give a class of functions f satisfying all conditions of lemma 40

67 Example Let A be a fixed, arbitrary, compact subset of the txspace E1 x En and let U(t, x) = U be some fixed, closed (but not necessarily bounded) subset of Em. Define fo(t, x, u) = g(t, x)h(u) + k(u) where g(t, x) is a continuous function on A, h(u) is a bounded, continuous function on U, and k(u) is a continuous function on U such that k(u) u I - +o as I ul -+ oo, u Uo Thus, f(t, x, u) is a uniformly continuous function of (t, x) in A with respect to u and f oul _ + oo as lul - +oo, u e U pointwise, and therefore by force of lemma 4, uniformly in Ao Lemma 5 will allow us to replace property (a) in Existence Theorem I by the weaker property {(3) instead of property (y) in the special case of the free problems of the calculus of variationso Actually, lemma 5 is slightly more general than needed for free problems. Its proof is due to Lo Turner [24]o Lemma 5~ Let A be a fixed compact subset of the tx-space E1 X En, and let U(t, x) - E for each (t, x) E Ao Let f (t, x, u) be a continuous scalar function on A x E and a convex function of m u for each (t, x) E Ao Then, fo lul + o as ul - + oo, u e Em pointwise in A if and only if the same limit holds uniformly in Ao PROOF~ The sufficiency of the condition is trivial.

68 Let us prove its necessity. Suppose flul 1 + oo as I ul -+ oo pointwise in A, but not uniformly. Then, there exists some N > O and a sequence (t x, x, UV) e A x Em, v= 1, 2,. such that f (t,xv, u )lu I- < N, v= 1,2,... and Iu -+oo OV V V V as v- + o. Without loss of generality, suppose (t, xv, u uVl 1), v= 1,2,..., is convergent to (to, x0, uo) e Ax E with I uo = 1o Now fo(to,x,u)lul 1 + oo as lul - +oo, ue U. SinceA is compact, fo(t, x, o) is bounded in A, say fo(t, x, o) I < C, there is a constant Xo such that X f (to xo uo) > N+C when X > X 000 0 - 0 (one can clearly assume that X > 1). By continuity, there is a a > O such that Xo' f (tX, u) > N + C if 0 0 0 (t,x) e N6(t,Xo) u,uo Em, Iu-u0o < 6. Let v be large enough so that (t, x ) e N6 (to, xo), i uo-uV u'- I < 5 and u I > X for each v > v o Then Xu Iu I = (1-X I u V | )o +x luo UV aio f v > v o v is a convex combination for v> v and fo(t, x, Xo u Iu l ) < (1-Xol uV I)fo(t, x, o) +XO luv f (t x u) < C +X lu I -1f tx, u for v> v. ~T~Qv0he V Vrefore Therefore

69 fo (tx -u)lu 1' > X-f (t, x, oU lu -1u) CX-1 02 -' 0/ ~ 0 V 0 > X0of (t,x,Xo u I1 ) -C -0 0 V'V 0 V as X can be assumed to be > lo Thus, f (t, xv,u)lu 1 > N which is a contradiction and lemma 5 is proveno Lemmas 3, 4 and 5 have immediate consequences when combined with Existence Theorem I. We give below Existence Theorem II, which is an equivalent form of Existence Theorem I, together with a few corollaries to Existence Theorem 11o When A is not compact but closed, condition (y) needs to be slightly altered to the new condition (y) defined by (Y)m 11 (x u if( ) - oo as io, u U(t,x), uniformly for (t, x) in any compact part A of A. For every compact part A of A there are constants C 0 D > 0 such that f(t, x, u) i < C + Do uj for each (t, x, u) e M with (t, x) e Ao - 0 0 0 Existence Theorem I1 (Equivalent form of Existence Theorem I)o Theorem I is the same as Existence Theorem I where the growth condition (a) is repla ced by (y) if A is compact. If A is not compact but closed and contained in a slab [t < t < T, x E ], to T finite, then Existence Theorem II still holds if (a), (b) hold, and (a) is replaced by (y)mo If A is not compact nor contained in a slab as above, but A is closed, then

70 Theorem II still holds if (a), (b), (c) hold and (a) is replaced by (y)m. Finally, condition (a) can be replaced in any case by condition (d)o This theorem is a consequence of Existence Theorem I and lemma 3, Corollary 1: This is the same as Existence Theorem II where for each (t, x) e A U(t, x) = U is a fixed, closed subset of the u-space Em, condition (y) and ()m are replaced by the assumptions that (i) fo(t, x, u) is a uniformly continuous function of (t, x) in A with respect to u, and (ii) f ul 1 -+oo as ul + oo, u e U pointwise in A It is clear that if U is a fixed, closed subset of Em, then U satisfies property (U) in Ao PROOFo This corollary follows from Existence Theorem II and lemma 4. Corollary 2: This is the same as TheoremII where for each (t,x) e A U(t, x) Em is the whole u-space Em, condition (y) and ()m are replaced by the assumptions (i) fo(t, x, u) is a convex function of u for each (t, x) e A, and (ii) fo iul - +oo as I ul -+ o, u E pointwise in A. (It is clear that U Em satisfies property (U) in A ) i In addition, if m = n and f.(t, u, il) u i 2. o o, n, then the condition that Q(t, x) satisfies property (Q) in A can be relaxed in the above case.

71 PROOF: The first statement follows from Theorem II and lemma 5. The last statement follows from lemma 5 of Chapter II and lemma 7 of Chapter III. Theorem II extends to Lagrange problems and problems of optimal control the analogous Existence Theorem II for free problemas proved by Tonelli in [23a] for n=l and f of class C, and proved by Lo Turner in [24] for any n > 1 and f of class C~0 Al~ o0 though condition (a) appears stronger than that given by Tonelli or Lo Turner, lemma 5 shows that it is not stronger in the case of free problems. Let us give an example of an optimal control problem to which Existence Theorems I and II apply, Example 1: Let m n 1, A [0, 1], for each (t, x) e A let U(t, x)- E1, let f(t, x, u) = u, f (t, x, u) u and boundary condi0 tions x(O) x(1) 0 0. Then U(t, x) satisfies property (U) in A and Q(t, x) Q {(zo, z)I z >, z e E1} is a fixed closed, convex subset of E1 x E1 and obviously satisfies property (Q) in Ao Now fo/u u2/ ul - l ul- + o as l ul -+oo, u E1 and the limit is uniform in Ao It is clear that Existence Theorem II applies. If one chooses, (z) - z for each z e [0, + oo), then f (t, x, u) - u > ul2 =(l ul) for each (t,x,u) E [0, 1]2xE1 and (z)/z-+oo as z - + co, z E E1, z > Q. Thus, Existence Theorem I also applies to this example as the equivalence of Existence Theorems I and II implyo

72 However, Existence Theorems I and II do not apply to the following example. Example 2: Let m n 1, A = [0, 1], for each (t,x) e A let U(t, x) - E1, let f(t, x, u) u, fo(t, x, u) tu and let the boundary conditions be x(0) = x(1) 0. Then U(t, x) satisfies property (U) for each (t, x) E A and Q(t, x) = {(zo z) z0 > tz 2 z c E} Now, Q(o x,6) {(z z) z > 0, z c E} (o, x). Hence, Q(o, x) n cl co Q(o,x, ) for each x e [0, 1]o Therefore Q(o, x) satisfies property (Q) for each (t, x) e {0} x [0, 1]. Since Q(t, x) obviously satisfies property (Q) for each (t, x) e (0, 1] x [0, 1], we conclude that Q (t, x) satisfies property (Q) in A. Although fo/lul -+0 oas lul -+co, u E1 uniformly on each compact subset of (0, 1] x [0, ], f (t, x,u) Iul does not approach + oo as I ul + oo for any (t, x) e A with t = 0. Hence fo/l ul does not tend to + ox as ul + oo, u ( E1 uniformly on A and neither Existence Theorem I nor II applies to this example. Nevertheless this example possesses an obvious optimal solution given by x(t) 0, u(t) - 0 0 < t < 1. Let us consider the same example with slightly modified boundary condition x(O) - 1, x(l) - Oo Theorems I and II still do not apply and the new problem is known to have no optimal solution. Indeed let i - inf I[x, u], where the infimum is taken over the class Q2 of all

73 admissible pairs, that is the class of all pairs x(t), u(t) such that x(t), 0 < t < 1 is an AC scalar function with x(0) 1, x(1) 7 0, with tx'2 L-integrable in [09 1] and u(t) x'(t) almost everywhere. Then lx, u] > 0 in SW and therefore i > 0. If we consider the sequence xk(t), uk(t), 0 t 1, defined by xk(t) - 1, uk(t) -0 for o < t,< k1 xkt) - - (log t)(log k)' uk(t) - - t log k)1 for k < t < 1, and k2, 3. o we have I[xk uk] (log k)1 and therefore I[xk uk] -0 as k - + co Thus, i-O. But I[x, u] - 0 implies that t u- 0 almost everywhere, and hence u(t) - 0 almost everywhere and x(t):- constant on [0, 1]. This is impossible as x(0) = 1, x(l) - 0. Thus, the problem above with boundary condition x(0) - 1, x(l) = 0 has no optimal solution and Theorems I and II do not apply to either of these examples. 12. An existence theorem with exceptional points. For free problems it was shown by Tonelli [23a] that the growth condition (a) can be dispensed with at the points (t, x) of an exceptional subset E of A provided some additional mild hypothesis is satisfied at the points of E, or E is a suitable "slender" set. This situation recurs for Lagrange problems as we shall state in Theorems III and IV below. Let us consider again general Lagrange problems (optimal control problems) as stated in No, 1, and let E be a given subset

74 of Ao We shall need a new condition, say (y*), another modification of condition (y) of noo 11o (y*) I ul -f (t, x,u) - oo as lul -+o, u E U(t,x) uniformly for (t, x) in any compact part A of A-E; for every compact part A of A-E there are constants Co, D > 0 such that If(t, x, u) < Co + Do u for all (t, x) e A, u E E(t, x). We shall need also a local property concerning the relative behavior of fo and f in the neighborhood of given points (to0 x ) e A. We shall denote it property (T), since it is modeled on an analogous condition used by Tonelli [23a] for free problems, n=l and f of class C o We shall denote by No (t, x) the open neighborhood of radius 6 of (t, x) in A, that is, the set of all points (t', x') e A at a distance < 6 from (t, x)o A point (t, xo) E A is said to possess property (T) provided there is a neighborhood N6 (to, Xo) in A, two functions q(), o 0, a > 0, / real, C > 0, with 0(Q) nonnegative, O(0 +) + co, i integrable in (0, &), () nonnegative, nondecreasing such that 5 4I)(K( 5))- + oo as 0 - + and such that (t, x) c N0 (to x ), u ( U(t, x) implies that

75 fo(t,x,u) > {t-t |a|ul ( pu, (32) j f(t, x, u) I< c + Do ul for each (t, x) E N (to x ), u e U(t,x) For instance condition (T) is certainly satisfied at the point (t0o x) c A if there is a neighborhood N6 (to, x0) in A, and constants k > 0, a >, c> 0, i real, C > 0, D > 0, such that f (t,x,u) > kit-t a ]uil +a+ o^~~~~~ 0 ~~~~~~~(33) f(t, x, ) I< C o Do ui for each (t, x) e N (to xo) u e U(t x o Indeed we have only to take f = 1, choose a number 3 such that a(a + ) < 3 < 1, and select OW) -= 90 < t < 19 I() - kl /a /a 0 < C < + c Here f is nonnegative, (O0 +) + oo since 3 is greater than zero, f is integrable in (0, 1) since 3 is between zero and one, I is nonnegative, nondecreasing since a > 0, and qt({) b( ))'O- * j- f kl/a (-C )r/a 1 / 1a -'( 9)/a, k1/la ~1 -( + C)/a and ( q(-)~I()) ()) +co as 0+ since 1 - (a ~+ - /a < 0 On the other hand

76 fo(t,x,u) >kltto 1 lu I +a+ - - tolulu + (k u l )/a a + =t-tuo + t= - ltt tliull +a(lul)+ In the Existence Theorems III and IV below concerning Lagrange problems, viewed as optimal control problems, E is a closed subset of EX E. Although this condition was not explicitly stated in Tonell'Ps papers [23] concerning free problems, we shall show that E can always be assumed to be a closed set for free problems in each of the cases corresponding to Existence Theorems III and IV, and, therefore, Tonelli's Theorems satisfy this hypothesis in Theorems III and IV below. We shall prove this property of the set E after each of the theorems. Existence Theorem III. Theorem III is the same as Existence Theorem I, where a closed exceptional subset Eof A is given, condition (T) holds at every point of E and condition (a) is replaced by (>*) if A is compact. If A is not compact, but closed and contained in a slab [to t T, x En], to, T finite, then Theorem III still holds if (a) and (b) hold (and (a) is replaced by (y*) and condition (T) holds on the exceptional set E). If A is not compact, nor contained in a slab as above, but A is closed, then Theorem III still holds if (a), (b) and (c) hold (and (a) is replaced by (~y) and condition (T) holds on the exceptional set E)o Finally, condition (a) can be replaced in any case by condition (d).

77 PROOF: Let us prove the result for the case when A is compact. If A is compact, then E cA is compact, Because of property (T), to each point (t, x) e E there is an open neighborhood N0 (t, x) with the properties described in property (T)o Therefore, by the compactness of E, a finite number of the neighborhoods No (t, x), say A. = N0 (ti, xi), i=l, 2, o.,, and (ti, xi) e E for each i i, cover Eo Consider the setA A - A which is clearly a i=1 1 i~l compact subset of A - E. By property (T) in each No (ti, xi), and 1 by the uniform growth condition in the compact set A, there exists real numbers i', i=O, 1,.., 9, such that f (t, x, u) > ji. for each (t, x, u) e M with (t, x) e Ai, i=O, 1, 2, o o o We shall denote by "sub i" any of the elements defined by property (T) relative to (ti, xi), i=1, 2,., 9 f. Letting t = max( 1O.., u 1), one has f (t, x, u) > - Aj for each (t, x, u) e M. Therefore, t2 I[x,u] - fo(t,x,u)dt > -Di, t1 where D is the diameter of A and hence the infinum i of I[x, u] in ~ is finite, Let xk(t), uk(t), tlk < t < t2k be a sequence of admissible pairs such that I[xk, uk] - i as k + o. We may well assume that

78 i <I[xksuk] < i+k < i+1, k=1,2,.... Let us prove that Xk (t), tlk < t < tk, k- 1,2,.., are equiabsolutely continuous vector functions. For each k,denote by Ti the set {tl(txk(t))e Ai, tk <t <t2k }, i=0,...; k=l 2,.. On A the usual growth condition (a) holds, and therefore, by the same argument used in Theorems I and II, the vector functions xk(t) are equiabsolutely integrable on Tok; in other words, given k ok e > 0 there is a 6 6 (, Ao) >0 such that for any subset H of Tok of measure <6 we have ok o (Ho) f Ix'k(t) Idt < e 4- ( + 1)1, k=1,2,,.. Now, if H is an given subset of [ tlk t2k] with meas H <6, then the subset H - Hn Tk also has measure <6 and we have ok ok o (H Tok) I Xk (t) I dt < e 4 1 ( Q+ 1) 1 (34) Let C - max oI C1,.. ], D - max [1, D1, o, DQ], * = mmin [1, I 1,...,oo For each i=1, 2..,, let us determine a number j{, 0 < 1 <6i, small enough so that o 0i(z)dz < e 8-D- ( + 1)- (35) and so that

79 z i i(z)l i(i(z)) > 8 pD E -1 (36) for all z with 0 < z < i Also, again for each i=1 29 o 9 k let us choose a number, 0i (j3). Then.. a. a i a ia+ i g,),- -1i Yi ~ i iY i Lety y max [1, Y19.oy]. For each i-=12,..., and k=14 2, 0e let us divide the set Tik into four disjoint subsets Eikj, j1,2, 3, 4, as follows. Let tit(i) [ti-i. ti + fi], and let Eikl {t lt Tik, t c tii), t uk(t)l < i(lt-ti)} Zik2 = {t t E Tik9 t ~ ti(0i), t a Eikl}. Eik3 = {tit, Tik, t ti(ii), luk(t) <_ Yi} Eik4 - {tlt Tik' t ti(fi)9 t Eik3} InEikl we have uk(t)l < i.(lt-til) and hence, by force of (35), (H i U E k) S iuk(t)dt < E (Eikl) IUk(t)ldt i eil-l( i-l K E ( ikl) S.i(It-tib)dt K

80 Q 3 < k 2 S i.(z)dz < 2 o e 8- D (- +1)-1 i=l InEik2 we have Iuk(t)l > i.(lt-til), hence luk(t)l-1(l t-til) > 1. Now, I(I uk(t)l) > i(.i( l t-til), and then a -a a. -a. l u k(t)l luk(t)l. uk(t) i(lt-t (luk(t))i l((t-til)) a. +a. a. I t-tI uk(t)l OI l uk(t) )o 1 k 1 k [t-til i ( t-t il) t i (i( l t-til))] (39) where 0 < It-t.l < i3 since t c Eik2 and t=t. is obviously in Eiklo By (36) we have a. ai( a. a -1 It-{ i lt-til)\i (Oi(lt-til)) > 8M, De and (39) yields luk(t)l < (8 a* f e)lt-ti luk(t)l i k(t)l) By (32) we have then I uk(t) < (8-1 D 1)[fo(t xk(t), uk(t)) - i]

81 for every t e Eik2. and hence also -1 -1 -1 uk(t) < (8'. D e) [fo(t, xk(t), uk(t)) + i]. (40) In Eik3 we have luk(t)l < yi and hence since y - max[1, Y19.oo y)] we have also luk(t)l < y for t U Eik (41) i=l In Eik4 we have luk(t)i > Yi ltil > (lukt) ) > ti(yi) a. 1+a. a. f (t, x t-p, I(t {) f x(txk(t),Iuk(t)) -i > lt-til luk(t) 1ii luk(t)), and hence -aC -a. -a. ukt)l < [fo(t,xk(t),u k(t)) - /i] {t-til luk(t) i Ii lluk(t)l) -a. -a. -a. < [fo(txk(t)'uk(t))+;4i Vi 1 (i) and by force of (37) also for t e Eik4 uk(t) I < (8. D )[fot, xk(t)9 uk(t))'+. (42) For every measurable set H c [tlk, t2k] with meas H < 6 we have now

82 (H) x'k (t) I dt -(HnU Tk) f lx'k(t) dt i-=O < (H nTok) S I X(t) Idt + (H UTik) x'k(t) I dt i=l and by force of (34), also (H) S Jxk (t) Idt <e 4- (+1)- + (H1 UTi) lxk(t)ldt. We have now, almost everywhere in Tik Ix'k(t) - f(t,xk(t),uk(t)) 1 Ci + D luk(t)- 1 <C +Duk(t)I, and hence i=l (H) lxI (t) ldt <e 4- (+ 1)l +(HnUT )J(C + D lu (t)l)dt e 41(2+1) + C meas H + D(Hn UTik)uk(t)dt i-l < 4 (+1) + C meas H + +D (H nh E.) f luk(t)idt) jl i=l1

83 By force of (38), (40), (41) and (42) we have now (H) I Jxk(t) Idt< 4 (+1) +C meas H+D ok 4 1D (+1)1 + D (8, D ) (Hn U E k2) [f (txk(t),uk(t))+L ]dt + D (Hn D )(HE UE dt +D(HnU Erk) f dt il 3 +D (8 p1 D e)(H U E ) [4)f i(txk(t),uk(t)) + ]dt. Since fo + i > Ofor all t e [tlk t2k] we have (H) jxk(t) {dt < e 441 ( +1) )+ (C+ Dy) meas H + e 4 1 Q(+1)1 t tik +'2D (81i D. e 5 ) / [fo(txk(t)) + H ]dt lk S4e + (C+Dy) meas H + e (1+ + ) 21 E + (C D y)meas Ho If we take 6 min [6o, (C + Dy)l 21 e] then for every measurable set H c [tik9t2k] with meas H < 6 we have

84 (H) {x'k(t) dt < 2-1 -+ The vector functions xk(t), tlk < t < t2k, kal, 2,. are equiabsolutely continuous. If one recalls that for this case there is a constant M > 0 such that f (t, x, u) > - Mi for each (t, x, u) c M and the fact that xk(t), tlk < t < t2k, k=l, 2,.. are equiabsolutely continuous, then one may consider the function x (t) as in Existence Theorem I and the proof proceeds just as before. The proof is complete for the case that A is compact. If A is not compact, let us consider an arbitrary compact subset A c A, and its respective parts A n E and AA - An E The set A n E is a compact part of A on which condition (T) holds. Therefore, by the same reasoning as in the beginning of the proof of this theorem there is a pi > 0 such that f (t, x, u) > - for a11 (t, x) c Ao, u U(t, x) (where 0 may depend on Ao)o This fact, the assumptions given for the compact case of A, and the reasoning given for the case of A not compact in Theorem I reduces these cases to the compact case. The theorem is completely proveno REMARKS: lo There exists a function f (t, x, u) satisfying all conditions of Tonelli's Theorem II, but, for which the sets Q do not satisfy everywhere property (Q)o

85 Take m n -, A [0 < t< 1, -1 <x < i], f- u, U [-oo tu3 > 0 for all (t,x,u) c Ax Uo For t i 0, 0 f satisfies the growth condition. The set E {(t, x) it - 0, -1 < x < + 1} is the exceptional set, and condition (T) is satisfied at every point of E since f > I tl jui and condition (24) (Tonelli, Opere Scelte, po 216) is satisfied with to 0O k l a =- 1, a I 1, -= Oo The function fol = tu3!1 l tlui3 is obviously convex in u for each t, and the function fo2 = max [0, l-xu] is also convex in u for each x. Hence f (t, X9 ) U f01 + f02 is certainly convex in u for each (t, x) c A. Now we have Q(t,x) {(z,u)lz > fo(t,xu), -co 1, -oo < < +oo On the other hand, for 0 < 6 < 1, Q(0,6) {(z, u) iz0 > max [0, 1 -6 u], 0 max [0,1 + 6u, -9 < < <+ 0}o Hence (0, 6 ) e Q(0, ), (0, -6 ) Q(0, -6), and finally -1 -1 (0,o -), (0 -6 ) e Q(0, 0;6). As a consequence (0, 0) e co(oQ0, 0;5 ), (0 0) ) 1 cl co Q(0,0;5 ) while (0, 0) ~ (1(0, 0)o We have here

86 (0, 0) c co Q(0, 0; 6 )o 6 2o In the proof of Theorems I - IV we may disregard property (Q) at the points (t, x) of a subset of A of the form {(t, x) I (t, x) e A, t e {t}}, where {t} is a given set of linear measure zero. The proof is the same. A set of points t of linear measure zero is always disregarded in the proof. Corollary 1~ This is the same as Existence Theorem III where for each (t,x) e A U(tx) = U is a fixed, closed subset of the u-space Em, and the part of condition (y*) where f ul1 _ + o as u - + oo, u e U(t, x) uniformly on a compact subset A of A - E is replaced by the conditions (i) fo(t, x, u) is a uniformly continuous function of (t,x) in A and(ii) f ou11 - + as lul -+oo, ue U pointwise in A - E. PROOF~ This statement is a consequence of Theorem III and lemma 4o Corollary 2: This is the same as Existence Theorem III where for each (tx) e A U(t,x) E is the whole u-space Em, the function f0(t, x, u) is convex in u for each (t, x) A - E, fo ul -+c as ul - + o, u c E pointwise in A - E, and the uniform growth condition of fo(t, x, u) on each compact part A of A - E, given in (y*) is omitted.

87 In addition, if m n, f(t, x, u) u, fo (t x, u) is a convex function of u for each (t, x) e E, and hence for each (t, x) E A, then the conditions that Q(t, x) satisfies property (Q) in A and E is a closed set can be omitted if the exceptional set E is chosen to be the subset of A at which condition (T) holds and condition (a) does not holdo PROOF~ The control set U(t, x) - E for each (t, x) e A obviously satisfies property (U) in Ao On each compact subset A of A - E, the convexity of f in u for each (t, x) e Ao the pointwise growth of f l u -1 to + -o as u + oo, and lemma 5, guarantee the uniform growth off i u1i to +oo as I ui - +o and hence that the condition (y*) is satisfiedo Theorem IIl applies and the first part of the statement is proven. We shall first show that under the assumptions of the second part of the statement E is a closed subset of the tx-space E1 xE o If (t, x) e E - A, then (t, x) has a finite distance to the closed set Ao Thus, there is an open neighborhood of (t, x) which lies entirely in the set En - A and hence in En+1 Eo If (t, x) e A-E, then f0 is a convex function of u and fo u 1 + Oo as I u - + oo at (t, x)o Lemma 5 implies, under the assumption of the second part of the statement, that in some neighborhood of (t, x) e A - E, f ui U +o as u -+ oo uniformly. Thus for each point in this neighborhood of uniform growth one has f ul 1 + o as { ui - oo pointwise and therefore this neighborhood of uniform

88 growth lies entirely in En1 - E as at points of E condition (a) is not satisfied. One has that E + - E is an open set and that E is closed in E1 n+l~ We shall now show that under the assumptions in the second statement the hypothesis that Q(t, x) satisfies property (Q) in A can be omitted from the first part of the statement. We note that the proof of Theorewm III for A not compact is reduced by additional assumptions to the proof for the compact A case applied to some fixed compact subset A of A. In order to omit the condition that Q (t, x) satisfies property (Q) in A under the assumptions of the second part of the statement, this latter fact and remark 2 to Theorem III imply that it suffices to prove that, under the conditions of the second part of this statement, Q(t, x) satisfies property (Q) on any fixed compact subset A of A except for a set A' of the form {(t, x) i (t x) e A0, t E {t}} where {t} is a given set of linear measure zero. We shall show that under the assumptions of the second part of this statement that the set {t} can be taken to be finite. Consider a fixed compact subset A of A. Then f 1 ul + oo as l u cc> o at each point of the set B - A (A-E) and f is a convex function of u for each (t, x) E Bo 0 0 As each point (t, x) C B has a finite distance from the set Ao' E, 0 lemmas 5 and 7 imply that Q (t, x) satisfies property (Q) at each point of the set Bo Condition (T) holds at each point (t, x) e A n Eo 0

89 The neighborhoods given by condition (T) cover the compact set A n Eo Consider a finite subcover of A n E. Then there are a O O finite number of ti, i=l 2, o o Q, such that a. 1+a. a. foxt,x, u) > it-t uil llu il iul) +/ii for each (t, x, u) e N6 (ti x.) x E where ai, z.i and are given in condition (T). As i.(z) is a nondecreasing function on 0 < z < + oo and a i > 0, il 9 2, o o the function fo u + as - u1 - + at each point (t, x) e A n E except when t - t. for some i, i=1, 29 o,o Each point (t, x) e A n E with t 4 t. for any i=-1 29 o o a 9 {, has a finite distance from the closed set {(t, x) I t, x) e A n E, t = t. for some il, 2o, o }.o 0 Therefore lemmas 5 and 7 guarantee that Q (t, x) satisfies property (Q) at each point gt( x) e A n E with t 4 ti for any i=-l, 2 0,. Thus, under the assumptions of the second part of the statement, Q(t, x) satisfies property (Q) on A except for the set {(t x) I (t, x) e A0o tt for some i, i:l 2 o o o } and the set tl t=ti, i=i 2, o o o } is a finite set and clearly has linear measure zero. The second part of the statement is entirely proveno Example 3~ Theorem III applies to the following example to which neither Existence Theorem I nor II applieso Let m n 1 A - [0, 1] for each(t,x) A letU(t, x) e E, 24 let f(t, x, u) ~ u, f (t, x, u): t u and boundary conditions x(0) = a, letf~~x~u)u, 0

90 x(l) = b where a, b are fixed real numbers. Then U(t, x) satisfies property (U) for each (t, x) E [0, 1]2, Q(t,x) {(zo z) z > t u42 z u, u E1} {z z) iz > t z, z E } and Q(0,x,6) ) {(zo z)l z > 0, ze E1} = Q(0,x) which is a closed, Q(O' X, 6o - 1 convex set. Hence, Q(O, x) = n cl co Q(0, x, 6) for each 5 x e [0, 1]. Therefore, Q(0, x) satisfies property (Q) for each (t,x) e {O} X [0, ]. The set Q(t,x, 6) = {(z z) i z > (t 6)2 z z e E1} if t 0 and < 6 < Itlo The latter set is a closed, convex set in E. Hence, Q(t, x, 6 )= cl co Q(t, x, 6) if t 1 0 and 0 < 6 < I t. Therefore, Q(t,x) - =n (t, x, 6) = cl co Q(t, x, 6) 6 6 and Q(t, x) satisfies property (Q) for each (t,x) e (0, 1] x [0, 1] and also for each (t, x) e A. Although fo/lul -+oo as lul - + oo, u E1 uniformly on each compact subset (0, 1] x [0, 1], f (t, x, u)/l ul does not tend to + oo as ul - +oo, u e E1 for (t, x) e {0} x [0, 1]o As a result f/Ji ul does not tend to + oo as u -+ oo, ue E1 uniformly in A and both Existence Theorems I and II fail to apply. Yet Existence Theorem III applies. Let E = {0} x [0, 1] which is a closed subset of E20 Then fo/lul - +ooas Iul -+oo, ue E1 uniformly on any compact subset of A - Eo Also, condition (T) is satisfied at each point of E. Let (0, xo) be an arbitrary point in Eo Thus,

91 2 \4 t2 1+2+1,a i ul+aiu f0 (t x9 u) t u t tu > l tI l u' i ui where a - 2. Consider an arbitrary neighborhood of (0 x ), say 0 N(x0, x) Let - 1 a 1 2, 4(z) z- 34 for 0 < z <, z) z- for 0 < z < + co in this neighborhood, Then O(z) is a nonnegative integrable function on (0, 1] with q(0+) - + oo and I(z) is a norn negative, nondecreasing function on (0, + oo) such that z O(z) V(4(z)) 3/ 4 "3/8 -1/8. zoz' ~ z / z -/8 +o as z- O + It is also clear that fox(tjx u) X - tu2 > tl a i uI 1a ul) for each (tx) ( N (0, Xo UE E1 where, a,9, (z) and (z) have been chosen as indicated. Indeed, Ifl - ul < C + Do0ul for each (t x) e N (O, X), u E1 where C = 0 Do - l1 This analysis shows that condition (T) is satisfied at (0, x ) E Eo As (0, x0) is an arbitrary point of E, condition (T) is satisfied at each point of Eo Finally, Existence Theorem III applies to this example. Note that the present example verifies (33) at the points (0, x) withk.- 1, a = 2, cr, b= — 0, C o 0 Do 0 lo Note that Theorems I, II and III do not apply to the examples 1 and 2.

92 13. An existence theorem with a " slender" exceptional set. If the exceptional set E is suitably "slender", then property (T) at the points (to, x0) c E is not needed. Let E be any subset of A. Then for any subset H of the t-axis we shall denote by E (H) the set of points 5 of the x -axis, i=1, 2,.., n, defined as follows 1 1 n E (H) { i there exists a point (t, x l0.,x ) c E, with t H, xi - }. We shall denote by A* [Ei(H)] the one dimensional outer measure of E (H), and we shall require below that p*[E(H)] =0, i-l, 2 o o,., n, for every subset H of the t-axis of measure zero. For instance, any set E contained in countably many straight lines parallel to the t-axis, and to finitely many (nonparametric) curves xi - i(t), i-l, o, n, t' < t < t", pi(t) AC in [tI' t"], certainly possesses the property above. The property above was proposed and used by Lo Turner to extend to free problems in E results of Tonelli for free problems in E1. Existence Theorem IV If A is compact, Theorem IV is the same as Existence Theorem I where a closed exceptional set E is given, condition (a) is replaced by (y*), and (L1) I*[Ei(H)] 0, i:l, 2,o.., n, for every subset of the t-axis of measure zero;

93 (L2): for every (to, x ) E A there are numbers 6 - t (, x) > 0, y y(t x0) > 0, r- r(to x) real, b. -= bi(to, x) real, i-l, 2 oo o 9 n, such that n f (t, x, u) > r + bfi(t, x, u) + y 1 f(t, x, u) I i-l for each (t, x) E N6 (to, x ), u C U(t, x) (L3): for each compact subset A of A there is a constant 0 M > 0 such that o - f(t, x, u) > - M for each (t, x) g Ao u e U(t, x) If A is not compact, but closed and contained in a slab [t < t < T, x E E ], to,T, finite, then Theorem IV still holds if (a), (b) hold [and (a) is replaced by (y*), provided (L1), (L2) and (L3) hold]. If A is not compact, nor contained in a slab as above, but A is closed, then Theorem IV still holds if (a), (b) and (c) hold [and (a) is replaced by (y*) provided (L1), (L2), and (L3) hold]o Finally, condition (a) can be replaced in any case by condition (d)o PROOF~ Let us consider an arbitrary admissible pair x(t), u(t), t1 < t < t2 from the complete, nonempty admissible class 2 and its restriction to a subinterval of [tl2 t2], io eo x(t), u(t),

94 t1 t t where t < t < t' < t2 Denote by 2' the set of all such possible restrictions of admissible pairs from Qo Then Q2is contained in Q2' Let us assume that A is compact. We shall first prove that inf2, [x, u] is finite. Thus, infQ l[x u] is finite as the inequality inf2, l[x, u] < inf I[x, u] follows from 2 c 1'. As the set A is compact by (L3) there is a constant Mo > 0 such that f(t, x, u) > -Mo for each (t, x) e A, u e U(tx). Therefore, t I[x, u] 2 f(t, x(t), u(t))dt > - MoD for each (x, u) E 0' where t D = diameter of the set Ao Thus infQ, x, u] is finite and so is inffI[x, u]. Fix a real number M > 0, consider the set {(x, u) I (x, u) e', I[x, u] < M} and denote it by {(x, u)}M. Then, we shall prove that the vector functions x(t) from {(x, u)}M have uniformly bounded variation when A is compact. This last statement will be shown by finding two constants a real, b real and b > 0 such that M > I[x, u] > a + bBVI x(t) i for each (x, u) e 6' with I[x, u] < M. The last inequality is proven after a special partition of the txspace is constructed and the trajectory x(t), t'1 < t < t'2 is divided in a special way into subtrajectories xpk(t), t'lpk < t < t'2pk We shall now obtain this special partitionO Condition (L2) guarantees that to each point (t, x) e A there is a linear scalar function z(v) r + bo v where v - (vl, o., o v ), 1' n

95 n b (bl 0 0 0 bn) or z(v) r + Z bivi(rq bl1 o o all real), a i=l 1 number v> 0, and a constant Z > 0 such that f(t, x, u) > z(f(t, x, u)) + v I f(t, x, u), (43) or f > z(f) + v(f) for each (t, x) e N/3 -(to xo), u U(t, x), and hence for each (tx) E N -(to xo), u U(t,x) where N3 (t xO) - {t9xl t-t l <3, Ix - xl < 3 6 for i:l,2, 2o.n} 36O oo0 0 -O A finite number of the above open cubes, N3 6- (ti x.i) i=l1 2 oo i 1 cover A, because A is compact. Divide Enl into cubes Epq whose sides have length 6 > 0 by taking E - {(t,x)l (p-1)6 < t < p6, qi-l) < xi < qi6, i-19, 2, 0o, n} where p, ql. n e are integers, q=(ql,. e. qn), 6 < min(619.oo 6. ) and where 6 is chosen so small that each cube EPq which has 10 a nonempty intersection with the set A, and its 3 - 1 adjacent cubes9 are completely contained within one N3 (ti xi)9 i=l 9 i (This can be done as A is a compact set and is covered by the 3 (ti xi)' i19 o0 o io ) Hence, one can associate to each cube Epq, which has a nonempty intersection with A, a linear scalar function

96 n z(v) r + b bivi, v= (vl, o. vn) and number v> 0, such that i-l fo(t, x, u) > z(f(t, x, u)) + vi f(t, x, u) I for each (t, x, u) e M with (t, x) eEpq or any of the 3n+1 - 1 cubes which are adjacent to EPqo The t-coordinates of the vertices of these cubes define a partition of the t-axis. Thus, there are two integers po q0 such that each vertex of the previously mentioned cube has t-coordinates of the form p6 with pt < p < qo. For each to e [Po, q6 ] the set E({to}), i=l, o o n, has measure zero. Given 7] such that 0 < r < 6/2 for each i=l, 2,..., n the set Ei(4to}) can be covered by an open set F of linear measure less than rj. The cartesian product Fo {t } x F1 x F2 x.. x Fn of the set {t} and these open sets are open in the hyperplane H(to) - {(t, x) (t, x) e E n+ t } t Now, (H(t0) - F0) n A is a compact subset of A since it is the intersection of the closed set H(to) - F and the compact set Ao As E is a closed subset of the compact set A there is a p > 0 such that the set N Am g(H(to) - Fo) - {(t, x) i dist ((t, x), H(to)-Fo) < pnfi3 } where dist (b, B) is the distance between a point b En+1 and a set B in En 1 has an empty intersection with the closed set E. This last statement follows by the argument belowo Indeed, F is an open covering of the intersection E(t0) E n H(to).

97 Thus, H(t ) - F is a closed set, and must have a positive distance from the compact set E, since in the opposite case, the set H(tj )-Fo and hence (H(to) - Fo) n A would contain points of accumulation of E, and thus would contain points of E since E is closed, but this is impossible because Fo is an open cover of all points E(t ), that is of all points of E which are on the hyperplane H(t )o. Thus, for each point (to, x ) e (H(to) - Fo) n A there is a cube N (t, ) = {(t, x)t I t-t I < p, I xll < p for i-=1 n} and the open cubes N (t o x ) cover the compact set (H(t) - Fo) a A. There exists a finite subcover N (to xi.) ihl, 2,9.o for P (H(t F ) F n A We note that the compact set u N (tto x.) o 0o i= P covers (H(t ) - F) n A and has an empty intersection with E as o 0 it is contained in the closed set N (H(t ) - F ) which has an p +1 0 0o empty intersection with the set Eo As B u N (to xi) is a compact subset of En - E and i-.1I P hence EB B nf A is a compact subset of A - E as both B and A'o 0 o are compact sets of En+1 now f 0uj 1 - o as ul - oo u U(t x) A uniformly on B and there exists constants C(t ) > 0, D(t ) > 0 0 0 - 0 - such that I f(t, x9 u) i < C(t ) + D(t ) u i for each (t, x, u) E M with (t, x) c B by condition (?*)o If S is an open n-dimensional cube such that for each (t, x) e A, x ( S, then the complement I of the compact set Bo in [(t -p t +p)xS] is not only a bounded open set but it is the union of a finite number of disjoint open intervalso

98 Also, the sets I and I = I n A have the property that each of their projections on the x -axis, i=1,..., n has linear measure less than 71, as the following argument shows. Indeed, if (t, x) e I then I t-t 0 < p and (to, x) a H(to) - Fo since (t, x) would then belong to B and not to Io as above. Thus, for any (t, x) e I, we have (to, x) e Fo, and the projections on each of the x -axis have linear measure less than a. In this manner we have associated an open interval (to-p, to+p) with each to e [po6, q06 ]. By taking a finite subcovering, and then a suitable contraction of the corresponding intervals we define a partition P: p 6 to < t1 <.. < tR q6 of [po, q06 ] and it may be assumed without loss of generality that the points p6 for p < p < q are all used in this partition. Refine the previous partition of E1 into intervals Qaq by means of n+1 the hyperplanes t - tj, j=, o o R. Thus, the new intervals are of the form aq {(t,x)ltai < t<t, (qi- 1l)6<xiqi 6 i-=l.,in} a -1, aa 0 P R. q- (q1. *., qn) and ta - t1 < 6. Let z (v) be the a a-1 - 1 aq linear scalar function and v > 0, the real number associated with aq the cube of the former partition which contains Q q. Summarizing we have the two following partitions:

99 1o A subdivision of the slab [po6 <t < q6, xeE n] into hypercubes EP p - p o q 1 q (q q qi integers, of side length 6, with corresponding linear functions z (v) and real numbers vp > 0o pq pq 2. Given r7 > 0, there is a linear subdivision of the same slab into intervals Qaq [t t < t (qi-l)6 <x q] a-l4, 2,, o o R, whose edges along the x -axis, il1, 2, 0 0n, still have length 6, independent of 7, such that fo(t, x, ) > z (f(t,x, u)) + qf(t, x,u) - aq aq for each (t, x, u) E M with (t, x) c Qaq n A and for each a, acrl 2, 00 R. In addition each slab [ta < tt, x E ] S is divided into two disjoint sets B and I, each made up of finitely a a many intervals whose edge along the t-axis of total length < r.o There exist constants C > 0, D > 0 such that If(t, x, u) I < C + D I u for each (t, x, u) c M with (t, x) c B, and f Iul1 + oo as I ul -+ oo, u U(t, x) uniformly on B n Ao 0 a Thus, the set A t x i {(t, x) ( A, t < t < } BA u 1A a' a-I - a a a where BA B n A, IAI nA, BA n A - B n I n A - n Aa a h a a' a a a ar and I has projections on each of the x -axis of linear measure

100 less than o. Also the exceptional set E is contained in R U 1o Moreover, the constants v, b a C a and D are inde— eaaq aqq a a pendent of 7. If C max (C1<,.. 9 CR) and Do - max(D1. o DR), then I f(t, x, u) < Co + D ul for each (t, x, u) M with R A (t, x) e U B o We may clearly assume that D > 0 without loss a~l 0 a= a of generality. Let r max I r 1, b= maxib i, v= min v aq aq aq where the maximum and minimum are taken over all (a, q) for which Qaq has a nonempty intersection with Ao Also, take a real number N > 2b ( + 4fiT )Do Now, suppose x(t), t' < t < t is the AC trajectory from some admissible pair (x, u) e T'o Let x x(t), ta- < t < ta, be the part of x(t) t'1 < t < t'2 (if any) defined on [t, t ]. Divide x into more subtrajectories x o, x a a CY a ali''''aT as follows The first end point of x is x(t ) (or x(tl) if t < t' al a-1 1 a-I 1 < t ); the second end point is either the x-component of the first n+l point where (t, x (t)) leaves one of the 3 - 1 intervals in the sectionA { (t, x) i(t,x) ) A, t < t <t } adjacent to any one of a a -ithe at most 2+1 intervals containing (t 1 x(ta _)) or x(t ) if (t,x (t)) does not leave these 3n+1 - 1 intervals (or x(t'2) if to < t2 < tao The first point of x is the end point of x and the end point of x2 is either the x-component of first point of (t, x )) which leaves the 3n+ - 1 intervals adjacent to the at most

101 2n+1 intervals containing the end point of (t, x (t)) or x(t ) if al a (t, x (t)) does not leave these 3n+ - 1 intervals (or x(t'2) if ta < t < t ). Continuing in this manner, x x(t), t < < t 2 a a a- i - a is broken up into subtrajectories xk kl,..., To This process ak' a must terminate after a finite number of steps since each subtrajectory xak except the last has length > 6. Thus, the domain of Xak is an interval, say Aako ak ak0 Let ak be the set of all t in A where (t, x(t)) e I and let ak ak a A' be the complement of A in A Thus A A u A? ak ak ak ak ak ak Consider any (x, u) E {(x9 U)}mo Then I[x u] f 2 f (t, x(t), u(t)dt > t1 > (I ak) f [rak b kfi(t, x(t)k u(t)) a-l= k=1 i=1 + kf(t, x(t), u(t)) ]dt + ( Ak) Sf (t x(t) u(t))dt} > (44) T R a >-rID + {-bt ( Ak) f(t, x(t), u(t))dti a-:1 k-l + v( A a) fS f(tx(t), u(t) dt + (A'k) Sfo(t x(t) u(t))dt where D denotes the diameter of Ao

102 As f ul1 _ + 0 as ul - + oo, u e U(t, x) uniformly on A 4 each B, a=l, R, foIu- 1 -+oo, as ul - + oo, u E U(t, x) R A uniformly on U B o We also note that there exists constants a=1 a C >0, D >0 such that lf(t,x,u)l < C +D Dul for each (t,x,u) e M, (t,x) e U B o Then, for the N > 0 given prea=l viously there exists a Y > 0 such that f oul1 > N for ul > Y, R A u e U(t,x) and each (t, x) e U B o Let a D {sup[NIul-fo(t,x,u)]l(t,x,u) e M, lul< Y}. Since Nul - f < D forte t'ak' ue U(t,x), [uj Y, we conclude that f +D Nlul for allt e A'tk ueU(t,x). We have I[x, u] < -(r +l Il)D + T al- k=l R a + (A' k)Jl~fo~t, x(t), u(t))+]tdt> - (rA+1 )D+ + {-b(Ak) lf(t, x(t), u(t)) dt (45) a=1 k=1 a- k l + v(A ak) f if(t,x(t), u(t)) dt + N(A' k) f lu(t) dt} Since IfI < c + Dolul for all t E Al' with D > 0, we have lul > (ifi - C0)D1 for all(t,x,u) E Mwith t Ak o, (46)

103 The relations (45) and (46) yield I[x, u] > - (ri + C Dl + CD+ T R T + {- b(A k) f(t, x(t), u(t)) dt + v(A k)f f(t,x(t), u(t) dt + a=- k1l + ND-1 (Ak) S I f(t, x(t), u(t)) I dt}. (47) Now (Ak UA' k) f(t,x(t),u(t))dtl (A k) S f(t, x(t), u(t))dti - l(Ak) x(t)dtl < 2 6 in1 and therefore (Ak) ff(t,x(t) u(t))dtl - i(A k) f(t,x(t), u(t))dtl < (Aa k U A' k) f(t, x(t), u(t))dtl < 2 6 fin1 We have I (A k) ff(t, x(t), u(t))dtl < 26 iv f + I (At' k) f(t, x(t), u(t))dtl < 26 -n+ + (A ak) S f(t, x(t), u(t)) I dt (48) for each a 1, 2, o R; k 19 2,. o To Relations (47) and (48) yield

104 I[x,u] > - (r+ ID + CoDo )D+ T R a + I {-b(26ivi + (A'ak) S f(t,x(t), u(t)) dt) + a=1 k-l + v(Ak) f(t, x(t), u(t)) i dt + NDo1(A') I, f (t x (t), u(t)) I dt}o (49) If we let X. k (Ak) I f(t,x(t), u(t))ldt, X'ak (A' ak) S f(t, x(t), u(t)) I dt, T T R a R a A- 11 XakandX' I I ak a-= k=l a=1 k-l we have I[x,u] > -(r + Dl +CoD)+ A + ND + R a -b( 2E E (2 6-iT + X' ) - b(26/ vn+ + XaT) ca- k-l

105 But X = (Alk) S If(t, x(t), u(t)) I dt > (A S f i(t,x(t),u(t))Idt > > l (A' kS fi(t, x(t), u(t))dt{ > > 6 - > 6 -6 /2 = 6 /2 for each a - 1, 2,. o R; k=1,.., T - and for i=l, 2,..., n. Also, N > 2b(l + 4fini4)D o Therefore I[x, u] > - (r + I Dl +C D )D + vX + NDo X' R a -b( Z (4X'kn+1 X'k) ( ^ k^(ack ak a=1 kl1 R - b (26 n+T + X' a > > - (r + Dl +C D" )D - 26 bRv^T + vA + ND A'x R Ta -b(1+4 ruT Ak > aI k=l > - (r + IDI +C D 1)D - 26bbR/fij + v +ND 0 0 + [NDO - b(l + 4/ii4i)]x' > - (r + D + CoD')D - 26 bRfil + vX b(l + 4J) X' (51)

106 If we let v - min(v, b(l + 4fnT- ), we have from (51) I[x, u] > - (r + i+CD )D - 2 bR/n+l + (BV x(t) ). Hence, -1 BV{x(t)I < [I[x,u] +(r + iDI + CoDo )D + 2bRli] v But (x, u) e {(x, u)}M and so I[x, u] < M. Therefore the vector functions x(t) from {(x, u)}M have uniform bounded variation. The same holds when Q replaces /S' as Q c M'. Let us prove that the vector functions x(t), t'l < t < t' of the family {x(t), u(t)}M are equicontinuous. Suppose they are not equicontinuous. Then, there is an e > 0 such that for each nonnegative integer j there is a trajectory x = xj(t), tlj < t < t2j. _j __ __t2j j j and two points t'j, t'2j such that tlj < t' < t' < t x. is a trajectory from {x,u}M, xj(t'jl) - xj(t' j) > E, and 0 < t Vj - t j < j; letting uj denote the control corresponding to xj, I[xj, uj] < M. Without loss of generality we can assume that t'jl to, t j - to, xj(t'jl) - x j, (t j2) -x as j -+ co, and then necessarily Ix2 - xl > E. As the trajectories xj(t), tlj < t < t2j, of the family {x(t), u(t)}M have uniformly bounded variation by what was proven before, let us denote one such bound by UMO The sets Ei({t^}) have measure zero0 Hence, as before,

107 they can be covered by open sets of measure < r for arbitrary r > 0. If r7 is not less than e/2, choose r7 so that it is less than e/2. Let F = {t }x, Fl x,..., xF n; then Fo is an open set in the hyperplane H(t )o By the same analysis we used in forming the partition of En into intervals Qaq, there is a real number p > 0, two sets n+1 B and I such that the set I is the complement of B in o 0 o o (t - p, t + p) x S where S is an n-dimensional open sphere such that for each (t, x) e A, x e S, (to-p, to+p) x S = B u I n S, the set I0 is a finite union of disjoint open intervals, the sets I and o I n A have the property that each of its projections on the x -axis, i=1,..., n, has linear measure less than r7, there are constants C(to) > 0, D(t ) > 0 such that If(t, x, u) I < C(t0) + D(to) i u A B0 A, f 1-1I for each (t,x,u) E Mwith(t,x) E B B n A, ful +oo as ul -+ oo, u e U(t, x) uniformly on BA and E(to) =E n {t } 0 0 O x E is contained in I n o Let N> D(t ) 4e 1[M +1 + 12Z - r - bUM ] where r,6,b are 0 the constants defined above and Z = infQ, I[x, u]. Then, there is aY>0suchthatflul1 > N when (t, x, u) e M, (t,x) e Bo, ul > Yo Divide the set [t'L, t'j2] into three subsets Ajl, Aj2 and Aj3 in the following manner. Let j3

108 Ajl {tit e [ tjl tj2], x(t)) BoA uj(t) exists and u.(t) >Y}, A2 - x{tit [t' jl t'j2] (txj(t)) e BA t i Ajl} and Aj3 {tlt Ltj't'2] t i Ajl U Aj2} Then the set [t jl t j2] Ajl U Aj2 U Aj3A Clearly there exists a real number j > 0 such that for j > j each t e [tvj, t' 2 satisfies the inequality I t-t I < p. We now have, for j > j I[x, uj] _ f J fo(t, xj(t),uj(t))dt 2 f(t, xjt),uj*t))dt t t tj2 + j fo(t, x(t(t), uj(t))dt >2Z+ f o(t, xj(t), u u(t))dt j2 tj ji > 2Z +(A f(t, xjt), uj(t))dt k- 2 by the growth condition of fo and the inequality fo > -r-bl f{ for 3x M o o >2each f(t, x, u) ( M as proven above As U A. is contained in k=2

109 3 U Aik =[t'j t' 2] we obtain for j > j k=1' I[xj, u] > 2Z - r(t2 - tjl) - bUM + N(Aj) u(t) dt > 2Z - rj - bUM + N(Aj) u(t) dt > 2Z - r- bUM + N(Ajl) uj(t) dt. Now (A 2)f uj(t) dt < Y(t'j2 - t'jl) < Yj Also, letting CO - C(to), Do - D(to) and assuming that Do > O (there is no loss in generality), we have uj(t)! > lf(t,xj(t), u(t))lD - CoDo for each (t, x) e B A0 u E U(t, x). In addition 2 2 ( U A k) S f(t, xj(t), uj(t))dt > ( u Ak) fi(t,xj(t),u(t)) dt k-2 - k 2 > ( u Ajk) f fi(t, xj(t), uj(t))dtl 3 u Ajk) fi(tU xj(t), uj(t))dt - (Aj3) ffi(t, xj(t) uj(t))dtl k=:Jl 3 {( u Ajk) S fi(t, xj(t), uj(t))dtl- I(Aj3) f (t, xj(t) uj(t))dt > Ix(tj2) - j(tjl) - 7I > - >> /2 - e/2 as n < E/2, for each i=l, 2 o..,n. Thus,

110 2 (A) j I u (t) dt = ( u A) j uj (t) dt - () u/ (t)idt -1 -1 -1 U( U A^) j lf(tx(t) u (t))l dt-(CD Y)j-1 > D E/2 - D c/4 D o E /4 by the above inequalities and for each j > j - max (Jo D 4e e -m 0 ^o -1 (CD- 0-+ Y)). But this last inequality implies that I[xj u] > 2Z - r - bUM +N(Aj1) J lu(t) dt > 2Z - r - bUM NDO (/4 for j > j Hence, I[xju.] > 2Z -r- bUM + (M+1) +12Z - r- bU I >M +1 > M for j > j as N > D 4' lM +1 + 12Z - r -bUM ] This is a contradiction. Therefore, the vector functions x(t), t'1 < t < t 2 of the family {(x, u)}M are equicontinuous when A is compacto Hence, these same vector functions are equicontinuous when A is compact if Q2 replaces Q2 8.

ll Now, let xk(t), uk(t), tlk t < t2k, be a sequence of admissible pairs, each from fS such that I[xk, Uk] - i as k -+ o. We may assume that t2k i< I[xk uk] fo(t, xk(t), uk(t))dt < i+l /k k-1, 2,.o tlk Therefore, the previous result implies that {xk(t)} are equicontinuous as I[xk uk] < i + 1/k < i +1 for k=l, 2.... As xk(t), uk(t), tlk < t < t2k9 is an admissible pair from 2 and A is a compact subset of E n There exists a constant A such that Ixk(t) < A for t [tlktk], andk=1,2, 00. Thus, there is a continuous function x (t), t1 < t < t2 such that p(xk, xo) - as k- + oo. xo(t), t1 < t < t2 is not only a continuous vector function but is also a BV vector functiono For each e > 0 there exists an Q and a partition t t' < t' < * < t' t2 such that either 1 1-2 2 Q -1 I xo(t' i+)-Xo(t'i)i > BVlx (t) li=l or i i-

112 accordingly as BV ixo(t) is finite or noto If we choose k~ so large that k > k implies that p(xk, x ) < e, we have that either Q -1 BVIx (t) - e < lxkt' il ) - Xktil +2(-1) i-1 < BV xk(t)I + 2(-1)e fork > k or Q -1 E lXk(t i+l) - xk(ti)I + 2(-1)e i-l <BV Ixk(t)I + 2(-1)e for k > k where xk(t) is xk(t), t1 < t < t2, extended to (-oo, +oo) by constancy. Since BV Ixk(t) < Ui1 < + oo the second alternative is contradictory for e sufficiently small. As a result BVI xo(t) I < + o and xo(t) is a measure induced on [tl, t2]. We shall now show that x (t), t1 < t < t2 is not only a continuous BV vector function but that it is also an AC vector function. Suppose it is not an AC vector function, then, there exists a Borel measurable set B c [t1, t ] which has zero Lebesque measure, but positive x0 measure, io e. x (B) > 00 Now, every set Ei(B), i:l, 2,., n, has zero Lebesque measure. Thus if E~ -- {(t,x)It e B, x xo(t), t1 < t < t}, the set [B E(B) x E(B). E(B)] has proectons of zero F; [B i, E (B) x E (B) x.., x Er(B)] ha~s projections of zero

113 measure on each x -axis, i=l, 2, o0 00 no Then, there is a closed set B'c B such thatx X(B) > 3L/4 where L - x (B) cnd ~ o ~ X~X0 t, xc(t)) i [B x EiB) xa o o X ED(B)j for each t z B 0 Now P = {(ex) t es B x = x (t)} is a compact subset of A-E and hence the set P n E is emptyQ Thus, th?,aere are two d.ujoint open sets 01 and 02 such that P' c 01 and E c 0.o Consequently A n (En 1 - 0 ) is a compact subset of A-E and contains P'. Indeed, there is a po > 0 such that the set N (P9) {(t, x dist ((t, x), P) < po} is contained in Po 0 -1 A n (En1 - 02). For N > 4D[i il +2 rD +bU i+]L where i - inf I[xx, u], there is a Y > 0 such that f (t. X ) > Nlul for each (t, x) A An (E+1 - 02), u J U(t, x) and hence for each (t, x) ( N (P'), u f U(t, x)o Let C, D0 > 0 denote the constants (and Do can clearly be assumed to be greater than 0) such that Ifl < Co Do i u for each (t, x, u) E M wilth (t,x) E A n (E + -02) and hence fcr each (t, x) E N (P'), u T U(t, x). We may assume without loss of generality that Y > C. Since B' is compact, there are infinitely many open disjoint intervals (a. /3.) j1 r, such r r that B' is contained in u (a., x., ( U (a,.)) < x (By) + =J J 0i 0 -\ 1r L 8 Y and a tt x (t)) marps U (a..) t irto N (PI). We ca n clearly assume without loss of generality thlt

114 r (j. - a) < min(L8 1 1Do p o/2) j-1 The following argument shows this. Now B' is a compact set with i*(B') 0. For each e > 0 there exists a bounded open set'0 which contains B' and such that l*(O' ) < e. Choose o < max(L8 Y Dop )p As O' is a bounded open set it consists of a finite or denumerable o number of open subintervals, say Oj, j=l, 2,... These subintervals O0 cover the compact set B' and hence a finite number of them also cover B'. Denote this finite subcover by 0., j=l, 2,..., j o Form the open intervals (a 1' 1) n 01,... (a, Or) n 01l * * A (a0, l) n 0j.., These intervals have the required properties if the original intervals (al, 1).. (a r, Or) do not. Then, we choose k so large that r E((ax j,) n [tlk t2k]) f(txk(t), uk(t))l dt > L/2 j-l r and (t, xk(t)) e N (P') for t [ u (a, j)] n [tlk, tk and all k > kop Thus k > k~. Thus

115 jl (( ) n [t u)dt j-1 3L8 D. r i,~ j( j^ e ) ik [k'tB {t x )k )t Bk { to te ~u [j,1j],, uk(t)l<YI and j-i r 1Bk u (-1 - j) - Bk Then (Bk) uk(t)l dt < YL/8YD L/8Do Hence-1 0 r LetB k{t: te ut a,{3], Iuk(t)I <Y} and r B - U (a., )-B Then k ji=l i k (Bk) f luk(t)Idt KYL/8YD L/8D Hence (B'k) S uk(t) dt > 3L/8Do - L/8Do L/4Do > 0 Therefore

116 I[xk, uk] > -rD-bUi1 + (Bk) f(t, xkt) ukt))dt > > -rD - bUi+ 1N(B'k) S uk(t) dt > -rD - bUi, - [4Do( il + 2 + rD + bUi 1)/L]L/4D > li +-2 > li +1 > i +1 and hence I[xk, Uk] > i + 1 which is a contradictiono Therefore x0(t), t1 < t < t2 is an AC vector functiono If we utilize the fact that (i) xk(t) - x (t) which is an AC 0,vector function in the p-metric and (ii) property (L3) holds, (L3) fo(t, x, u) >- M for each (t, x, u) e M for some M > 0 when A is compact, and if we apply the reasoning given in Existence Theorem I after the equi-AC of the sequence {xk(t)} is proven, then we obtain the result that there is a measurable control function such that x(t), u(t), t1 < t < t is an admissible pair from Q and I[xo uo] < i. As i < I[xo uo], one has that the absolute minimum if I[x, u] exists and is taken on in o Thus, the theorem is proven in the case that A is compact. When A is not compact, the reasoning given in Existence Theorem I can be applied with the various conditions stated in this theorem and utilizing property (L3) to reduce these cases to the case where A is compact. The theorem is completely proven.

117 Corollary 1: This is the same as Existence Theorem IV where for each (t, x) ( A U(t, x)> U is a fixed, closed subset of the u-space Em, and the part of condition (?^) where f 0 u! -1 + oo as I ut + oo, u ( U(t, x) uniformly on any compact subset Ao of A-E is replaced by the conditions (i) and (ii) of Corollary 1 to Theorem IIL PROOF~ This statement is a consequence of Theorem IV and lemma 4. Corollary 2: This is the same as Theorem IV where for each (t, x) e A U(t, x) = Em is the whole u-space Em, the function f (t, x, u) is convex in u for each (t, x) e A-E, f ul 1' + o as I u -+ oo, u E E pointwise in A-E, and the uniform growth m condition of f (t, x, u) on each compact part Ao of A-E given in (?*) is omitted. In addition for free problems, that is, when m - n, fi(t,, u) = u, i=1, 2, o o on, f (t, x, u) is a convex function of u for each (t, x) e, E, then the condition that Q(t, x) satisfy property (Q) in A and that E is a closed set can be omitted if E is assumed to be the minimal exceptional set (fo lu does not approach + oo as I u -+ oo at points of E) and fo is a normally convex function of u in Ao PROOF: The first part of the statement follows by the same argument of the first part of the statement of corollary 2 to

118 Theorem III. The statement that E is a closed set in the second part of the statement above follows by the reasoning used to prove the same property in corollary 2 to Theorem IIIo Q (t,x) satisfies property (Q) in A as f0 is normally convex in u (and hence quasi-normally convex), f=u, m=n and statement (viii) in Chapter I applies. 2 Example 4: Let m n-1, A= [0,1], for each (t,x) r A let U(t,x) - E, let f(txu) - u 2_2 f (t x u)x u -u foru >0, uc E1 022 x u for u >0o u EE and boundary condit.onsx(O) x(l) 0o Then, U(tx) satisfies property (U)'foreach (,x) e A and Q (tx) -{(z0, z)z > 22 22 x z + zfor z -0, z >x z for z. 0}o Now, Q(t, o,6) i(zo Z) z >z for z,0, and z'0 for z> 0} Q (t,o). Hence Q (t,o) = f cl co Q (to, 6) for eacht e [0, 1] Therefore Q (t,o) satisfies property (Q) for each (t,x) E [0, 1] x {0}o As Q (tx) also satisfies property (Q' for each (t,x) e [0,1] x (0, 1], we conclude that Q (tx) satisfies property (Q) in A. Although f Iul1 + oo as ul - + oo u e E uniformly on each compact subset of [0,1 ] XI (0, ], fo(t,x,u) lulI does not approach + o as lul + oo for any (t,x) c A with x-0 Hence, f lua does not tend to + oo as i[u - + oo, u e E uniformly on A and both Existence Theorem I

119 and I do not apply to this example. Indeed, even Existence Theorem III does not apply to this problem. Clearly, the closed exceptional set E for Theorem III must contain the set [0, 1] x {0} where the growth condition fails, but condition (T) is obviously not satisfied on this set. Therefore, Theorem III does not apply. However, Theorem IV applies. Let E [0, 1] x {0}. Then ju*[E(H)] - 0 for every subset of the t-axis of measure zero (in fact for any subset H c [0, l]) and (L1) holds. If r=0, bl S Y 1> 0 then f (t2 22 +~ u > 0 2-1 u + 2-1 - I U1 0 fo(t,x, u) - x u +u> +2 u+2 - ui > ufor u > 0, (t,x) A x2 2 "> u1 1ux= u >0~2 u + 2 iui =0 for u < 0, (t, x) e A and (L2) holds. Also9 fo(t, x, u) > 0 and (L3) holdso Now the growth condition on f (t, x, u) of property (y*) was shown to be valid on any compact part A of A-E, Letting C=0O D=1, if(t,x,u) ul 0 +1 I ul for each (t,x, u) e A E1 and (y*) is valid. Therefore Theorem IV applies, but Theorems I, II and III do noto REMARK Theorem IV is similar to a theorem of Tonelli [23a] for free problems, n=l, and f continuously differentiable. TonelliFs statement has been extended to free problems with n > 1

120 and f not necessarily differentiable by L. Turner [24]. The present theorem differs from that given by Tonelli and Lo Turner in two ways. Condition (L3) was not required, but normal convexity of fo(t, x, u) was required in place of condition (L2); hence one condition was added and one condition greatly weakened from the case of the free problems. Examples 1 and 2 are due to Tonelli,

Chapter III Optimal Control Problems where f is Linear 14e A few lemmas. We shall now consider the case where all functions f.(t, x, u), i=l1 2, o., n, are linear in u, and the control space is a fixed closed convex subset of E for each (t, x) of A. Precisely, we shall consider the optimal control problem t [x, u] S f (t, x, u)dt - minimum, (52) t1 m dx dt - g(t gjt, x) gt x) i=l 2, oo, n, (53) j=1 where x (x, o. X) ) Ex, and f (t x, u) is a convex function of u, u ( U, for each fixed (t, x) e A. If H(t, x) denotes the n x m matrix (gi.(t, x)), and h(t, x) the n-vector (g.(t, x)), then the differential sysI1 1 tem (53) becomes dx/dt - Ht, x)u + h(t, x)o We shall, henceforth, assume that g. (t, x), gi(t x), i 2,, o.., n; j=l, 2,9, m are continuous bounded functions on Ao The sets Q(t, x), Q(t, x) relative to the above problem are 121

122 Q(t, x) -[zz - H(t, x)u + h(t, x), u E U] C E (54) n Q(tx) - [z = (z z) z~ > fo(t,x,u), z H(t,x)u +h(t,x), u U] c E We shall need a few lemmas concerning the sets Q(t, x) and l(t, x). The proofs of the next two lemmas are due to Cesari [4a]. Lemma 6: Let the set A be a fixed closed subset of the txspace E1 x En, U(t, x) - U be a fixed, closed, convex subset of Em for each (t, x) e A, f (t, x, u), (t, x, u) e A x U, be a continuous scalar function on M = A x U which is also a convex function of u for each (t, x) e A, and the differential system be given by (53). Then, both sets Q(t, x) and (t, x), which are defined by (54), are convex for each (t, x) E A. PROOF o The set Q(t, x) is obviously convex for each (t, x) e Ao Let us now give the proof for the set Q((t, x). Let p =(p,p), q- (q,q) be any two points of (t,x), let 0 <a <1, o and z - (z, z) = ap + (1-a)q. Then for some vectors u, v e U we have p > fo(t,x,u), p - Hu+h P - 0 q > f(t, x, v), q Hv h, z = ap + (l-a)q, z -ap + (1-a)q, z = ap + (l-a)q. Now the vector w - au + (1-a)v E U as U is a convex subset of E.

123 We have z ap + (1-a)q -- aHu -- h) + (1-a)(Hv + h) = - H(au + (1-a)v) + h - hw, z - ap +1 (-a)ql > a f(t, x, u) + (I-a)f (t, v)> > f t, x, au + (1-a)v) fo(t, x, w). Thus, z (z, z) {E (t, x) and ~(t, x) is a convex set for each (t, x) ( Ao Lemma 7: Let the set A be a fixed closed subset of the tx-space E1 x En, U(t, x) = U be a fixed, closed, convex subset of Em for each (t, x) E A, f t, x, u), (t, x, u) e A x U, be a continuous scalar function in M - A x U which is also a convex function of u for each (t, x) e A, and for each compact part Ao of A let D b(z) be a continuous scalar function in the set Z - [z z = i ul for some u E U] such that f (t, x, u) > %~ (I ul) for each (t, x, u) e A x U and o(z) - + oo as z - + co, and let the differential system be given by (53). Then, the set Q(t, x), which is defined by (54), satisfies property (Q) in A. PROOF: We have to prove that Q (t, x) n cl co Q(t, x, 6 ) 6 It is enough to prove that n cl co Q (t, x, 5 ) c Q (t, x) as the opposate nclusion is trivia Let us assume that a given point z- (z ) el cl o Q (t, xs ) and let us prove that z (z z) z ( Q(t, x). For every 5 > 0 we have

124 z (z z): cl co Q(t, x, 6 )and thus, for every 6 > 0O there are points z z z) 5 co Q (t x, 6 ) at as small a distance as we want -0 from z, (z z). Thus, there is a sequence of points Zk. (z k zk) co Q(t; x5 k) and a sequence of numbers 6k > 0 such that 6k - 0 Zk z as k - oc, In other words, for every integer k, there are some pairs (t'k, x'k), (t"k) x" k) corresponding Zk k k-tk k^ k Z - z -a )k k k Z +k 1 -ak)zk, k -- kk (l-"kZk 9 k -k t k - l-kZk9 o k - ke k k k' k Z k tk x k > fOtk' Xk' k' k (t Xutk )Uk h(tj, X) 55) tk x xk tk9 xkX, Zk Zk Zk as k- + oo Because of these limit properties, there is some closed ball Bc E0l such that (t'k x') e B, (tk9 xk) ( B and (t, ) e B. Let A - B n A and consider 4> on this set. The second relation (55) shows that of the two numbers zk, Zk orne must be < Zk. It is not restrictive to assume that z < z for all ko Then the fourth relation (55) together with the JK "",K~~~~~~~~~'

125 lower bound for f yields 0 k> zk > fot'k9k k) > kIoU' k where zk z a.nd hence [Zk! is a bounded sequenceo Thus, lokI ul) < z k and the boundedness of the sequence [z k] together with this previous inequality and the limit property and continuity of o (z) implies that [(6:, uk l)] is also a bounded sequence. Finally, [u'k] is also a bounded sequence because of this last result and the growth property of o We can select a subsequence, say still [uT'k, which is convergent, say uk u' c U as k -+ o. The sequence [ak] is also boundedo hence we can select a further subsequence, which also we shall call [a k] for which ak a as k + oo with 0O a c< 1 Let uk U be the point k u a'k + (l-ak)u" (this can clearly be done as U is a convex set in Em). Then, Zk akZk (1-ak)Zk akH(t'k9 xk)uk + h(t k, x k)] + (1-ak)[H(t"k9 X"k)U"k + h(t"k9 x k)] t H(tk, xvk) [aku'k + (1-ak)u'k) + h(t/"k X"k) + ak {[H(tk9 xk) - H(t"k x"k) uk + + [h(t k Xk) - h(tt'k x'k)]} k- H )t"k x k h(t xk) Ak9 56)

126 0 o' O0" Zk - akk +(1-k)zk > > a kfo(t k' X'kI U'k) + (1-e k)fo(t"k, x"k U k) >- akf (t" k' X'"kt uk) + (1-ak)f (t" k X" k u'k) ak[fo(t k k9 kUk) - fo(t"k x'k, k)] > fo(t" k xks ak'U'k + (1-ak)U"T + +ak[fo(t'k' X'k9 Uk) - fo(t"k, X"k9 u2] >- fo(titk' x k' Uk) + Ak where Ak and Ak have the property Ak -0, AO - O and h(t"k,k) h(t, x) as k - + oo since t'k - t, x'k - x, t'k - t, X"k - x and'k -' as k - + oo. Again we can conclude that [b(IukI)] is a bounded sequence, and so is [Uk]. Hence, we can further select a convergent subsequence, say still [uk], with Uk -U U. Relations (56) now imply as k - + co z - H(t, x)u + h(t, x) -O -0 > fo-E, u )o Thus, z = (zo, z) ( ((,x), and lemma 7 is proven. The following example shows that Q (t, x) does not necessarily satisfy property (Q) for each possible optimal control problem. Lemma 7 does not apply to this case as f (t, x, u) fails to satisfy the "uniform growth" property implied by the function. b(z) such that

127 <4z)'-+oo as z - - Cc, z > 0 and f (t, Xu) > I ul) for each (+t x, u) s A x U. This example was given by Cesari [4a]. Example 5: Let m — n -= 1 U E1U A [0 1]2 fo(ttx u) - t3u zand f(t, x, u) tU Then, Q (t, x) - Q (t) - [z - z z) I z tu, u E1]. Thus, Q(0, x) - QO) - [(z, z) z >0, z - 0] and for t 0,,(t) z [(z z) z > tz, z e E1] and cl co Q(06 ) is the entire half plane [(z, z) Iz > 0, z c EJo Therefore (t, x) (- (t) does not satisfy property (Q) at t 0. However, there are optimal control problems for which the "uniform growth condition on f (t, x, u) can be relaxed and the set Q (t, x) still satisfies property (Q). The following lemma states such a case. But before we proceed we shall define a new property, c.ailled (SB), for the control set Uo (SB): A fixed control set U in Em is said to satisfy property (SB) if for each i, i=-, 2,. o m, there is a real number a. such that either u. < a. for each u e U or u. > a. for each u c Uo I - l i-i Examplesof such control sets are the sets U = [(u1,U2)lU > 0 > u2 > I] where a1 0, a2 land U = Ul U lu1 ) > 1, u2 < -11 where a1 1, a2 -lo U (UV 21/1 L 2 -1' 2

128 Lemma 8: Let the set A be a fixed closed subset of the txspace E1 x En, U(t, x) U be a fixed, closed, convex subject of the u-space E for each (t, x) e A which also satisfies property m (SB), let fo t, x, u) be a continuous scalar function'On M A x U which is also a convex function of u for each (t, x) e A, m n and f(t, x, u) - u. Then, the set Q(t, x), which is defined by (54), satisfies property (Q) in Ao PROOF: Proceed as in lemma 7 until relations (55) are reachedo The new relations (55) are as follows: Zk - k k + (1-ak)Z k O O? Ot" k Zk kzk + (1-ak)Zk, Zk = akZk +' (1-ak)Z"kP zk > f (tk x kk), Zk z - Uk >k V - o'k k k -k and such that t' -t xk -x, t x" - x, Zt - z-k- Z k Z zk -z as k- + oo As U is a convex set, z aku'k + (1-ak)uk ( U. Also sU Uas z k z as k- oo and U is a closed set. If we write the third relatiort of (57) in component form with the help of relat+iolns five and seven of (57), we have zk aku k - kl for k 2,

129 But zki) zo Now either there are only a finite number of ak - 0 or 1 or there are noto Suppose there are only a finite number of ak 0 or 1. Then nd u') and u' are both bounded or both k k unbounded as 0 ak < 1 for each k, ak> 0, 1-ak >0for eachk and their convex sum z i) - z() as k - oo where z(i) is some finite number. In fact, if u' and u" are both unbounded, they k k are unbounded with opposite signs. As U'k, u"k ( U, and U satisfies property (SB), this cannot happen. Therefore, both uki) k and u"k k i=l, 2,.., m, are both bounded and so are u' and u"k bounded m-vectorso Suppose there are an infinite number of k-values for which ak - 0 or ak k 1o Consider ak - 0 an infinite number of times. Then, zk - u" and zk z implies that k k U"k -z U for these values of k. Re -label this new subsequence with the same index k and we obtain new relations (57) with Uk - Z e U. For ak 1 an infinite number of times, the same logic yields new relations (57) for which u'k - z e U. Now, Zk = akUtk + (1-ak)u'k Ulk where uk = Cakuk (lak)Uk9 (58)

130 0 0o o" Zk - akZk + 1-la k - > a f!tl,XI UV + 1 -a )ftx~ (i T Ulf > -akf0{(t k Xk u k) + (1-ak)fo(t' k X' k U"?kk) + a kfo(tIk Xk, u k) - fo(tt k' x'k' U"k)] >0 fott"k 9 X"ks Uk) + k where A - akfo(t k' xk u) f(t?' xk )]. In the first k k0fWtkkk O k'k' k and third cases (58) is valid and uik approaches a finite limit, say u' as k - oo. Thus as t'k -t, t"k t x xk x - x and ak is bounded between 0 and 1, A k 0 as k - + o Hence, the relations (58) now imply as k - + oO. z > fo(t9, u). Thus, z (z z ) e Q (t, ), and lemma (8) is proven for the first and third cases. Consider the second case where (57) is valid and u"k and Uk approach a finite limit in U as k + cO. Again,

131 Zk akU k + (1-ak)u k - Uk o'? O T? - zk akk (1ak)k Z > a kfo(t k' xAk, U'k) + (l-ak)fo0(t k9x qk UV k) + (1-ak)[fo(t"k, X"k U"k) - fo(t' xk,', Uk)] Z > fo(t k'xk' k) + A where k kk k kk Thus, in where ak = (1-eak)[fo(t' k' X'k' Uu' k) - fo(t k9 x'ks'u k)]O Thus, in the secona case, with t' f, x' - t"k - t, x"k- x as k - +oo and O < 1 - a <1 for each k, A - 0 as k - + oo Hence the last k k relations imply that z u -o Therefore z - (z, z) c Q (f, x) as before and lemma 8 is completely proveno Example 5, which precedes this lemma, shows that property (SB) cannot be relaxed in lemma 8.

132 15. Existence theorems where f is linear in Uo It is now possible to state several existence theorems for optimal control problems in which the differential equation is linear in the control variable. Existence Theorem V: Let us consider the optimal control problem described in relations (52) and (53). Let the set A be a fixed compact subset of the tx-space E1 x Enr let U(t, x) - U be a fixed, closed, convex subset of the u-space E for each (t, x') A, let f (t, x, u), (t9 x, u) e A X U, be a continuous scalar function on M A x U, which is also a convex function of u for each (t, x) E Ao Let 1 09 0 < < - r9 be a given continuous function of ~ such that (C )/ + ooo as - oo and f(t, x, u) > I ul) for each (t, x, u) e A x Uo Let 2 be the class of all pairs x(t), u(t), t t < t29 x(t) absolutely continuous, u(t) measurable satisfying (56) ao e.. Then, the optimal control problem has an optimal solution. If A is not compact, but closed and contained in a slab [ t < t < T, x. E 1 tr T finite, then the result still holds if a..... condition (b) holds, If A is closed, but not contained in a slab as above, then Theorem I stll holds if conditions (b) and (c) hold.: (c') foi (t X, u) > > 0 for each (t, x u) A x U where J is a positive constanto

133 Finally, for A not compact, the condition f > (lul) can be replaced by conditions {d) and (e) (e) for every compact subset A of A there is a function 4 as above such that f )> j (I u) for ea.ch (t, x, ) e A x U{ PROOF: By lemmas 6 and 7 of this chapter, the set ((t, x) is convex for each (t, x) ( A and satisfies property (Q) in A, where A is a fixed closed subset of E1 x Eno The set U is a fixed, closed, convex subset of Em for each (t, x) e A and obviously satisfies property (U) in A for each caseo Now gij and gi are bounded, continuous functions of (t, x) on A and hence there is a C > O such that I g. < Co I gi < C for each (t, x) ( A and fl < IHu + hl < I HI lul + ihl < nC lul + nCc for each (t, x, u) ~ A x U for each case (A compact or A not compact), The case for A compact is now just a special case of Theorem I from Chapter II The case for A not compact, but contained in a slab is also proven if condition (d) is satisfied, io eO. if f (t, x, u) > E i f(t, x, u) for all (t, x, u) e A x U with xi > F and for some constants E > 0, F > O Let us prove that condtion (d) is satisfiedo Indeed, <C)/{ > 1 for all Id1 > D for some constant D> 0O Then, for Iul > D we have Iu t <K IluI) aid hence Iul < D +-E ul) for each

134 u e U. Now for each t, x, u) A x U we have fi I- Hu + hl < IHI lul + ihi < iHI(D +(iul)) + jhl <: IH ll u1) +(D HI + lhl) < [IHI +(DiHI + lhl)1 ](SuI)as(lui)> > 0, < [IHI +(DIHl + IhI)ll]f Thusf > [IHI +(DIHI + h['-l!] fl > Elf for some constant E > 0 and for each (t, x, u) e Ax U as IH and Ihi are bounded on A since gij. gi are continuous bounded functions on A. Thus, the case where A is not compact, but contained in a slab is proven. Suppose A is closed, not compact, nor contained in a slab as before. Then, the conditions of Theorem I in Chapter II are verifiedo This case is also proveno The last statement follows as Theorem I from Chapter II only required this to hold, and neither lemma 6 nor lemma 7 require more than condition (e). The theorem is proven. Existence Theorem VI: Let us consider the optimal control problem described in rela.tions (52) and (53)o Let the set A be a fixed compact subset of the tx-space E1 > En let U(t, x) U be a fixed, closed, convex subset of the u-space Em for each (t, x), A, let

135 f (t, x, u)9 (tt, x u) - A >< U, be a continuous scalar function on M A x U which is a convex function of u for each (t, x) e A and is such that u1 fo(tx, u) - o as i u - + o u U uniformly in Ao Let 2 be the class of all pairs x(t), u(t), t < t < t2, x(t) absolutely continuous9 u(t) measurable satisfying (53) a. eo. Then the optimal control problem has an optimal solution. If A is not compact, but closed and contained in a slab [to < t < T, x E En], to, T, finite, then the result still holds if condition (b) holds. If A is closed, but not compact, nor contained in a slab as above, then Theorem VI still holds if (b) and (c) hold. Finally, for A not compact, the condition f l ul + oo as ul - + oo, u f U uniformly on A can be replaced by conditions (y) and (d) PROOF~ The proof follows as a result of the reasoning given in Theorem I of Chapter III and lemmas 3 and 7 in Chapters II and III, respectively. Existence Theorem VIlo This theorem is the same as theorem V, where a closed exceptional subset E of A is given, condition (T) holds at each point of E, and conditionr (a) is replaced by (y*) if A is compact (the inequality on f in condition (V*) is obviously satisfied in this case)~

136 If A is not compact, but closed and contained in a slab [t < t < T, x En], to, T, finite, then theorem VII still holds if conditions (b) and (d) hold (and (a) is replaced by (y*) and condition (T) holds on the exceptional set E). If A is not compact, nor contained in a slab as above, but A is closed, then theorem VII still holds if conditions (b), (c, and (d) hold. PROOF: This theorem follows from Theorem III by the reasoning contained in the proofs of Theorem V and corollary 2 of Theorem IIIo Existence Theorem VIII: This theorem is the same as Theorem V, where a closed exceptional set E is given, condition (a) is replaced by (yS*), conditions (L1), (L2), and (L3) hold, and E is a set on which condition (a) is not satisfied (the inequality on f in condition (X/*) is obviously satisfied in this case). We shall also assume that ((t, x) satisfies property (Q) on the set E. If A is not compact, but closed and contained in a slab [to t < T, x En] to, T, finite, then Theorem VIII still holds if conditions (b) and (d) hold (and (a) is replaced by (y*), provided (Li, (L2) and (L3) hold).

137 If A is not compact, nor contained in a slab as above, but A is closed, then theorem VIII still holds if condition (b), (c') and (d) hold (and (a) is replaced by (y*), provided (L1), (L2), and (L3) hold). In addition, if m=n, f(t, x, u) - u and the set U satisfies property (SB) then the condition that Q(t, x) satisfies property (Q) on E can be omitted in each of the above cases. PROOF The first statements of this theorem follow from Theorem IV and the reasonings contained in Theorem V and corollary 2 to Theorem IV. An application of lemma 8 proves the last statement.

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