THE UNIVERS I T Y OF M I C H I GA N COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Progress Report No. 18 EXISTENCE THEOREMS FOR WEAK AND USUAL OPTIMAL SOLUTIONS IN LAGRANGE PROBLEMS WITH UNILATERAL CONSTRAINTS >I,:i"xi stence Theorems.o. o~..., La mbero Ce.sati',Oi.RA iro ject 07100 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. GP-3920 WASHINGTON, D.C. administered through: OFFICE OF IESEARCH ADMINISTRATION ANN ARBOR September 1965

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EXISTENCE THEOREMS FOR WEAK AND USUAL OPTIMAL SOLUTIONS IN LAGRANGE PROBLEMS WITH UNILATERAL CONSTRAINTS II. EXISTENCE THEOREMS* Lamberto Cesari Department of Mathematics The University of Michigan Ann Arbor, Michigan In the present paper (II) we prove existence theorems for weak and usual optimal solutions of nonparametric Lagrange problems with (or without) unilateral constraints. We shall consider arbitrary pairs x(t), u(t) of vector functions, u(t) measurable with values-in Em, x(t) absolutely continuous with values in En, and we discuss the existence of the absolute minimum of a functional t2 I[x,ul _ Z:fo(t,xit), u(t))dt, with side conditions represented by a differential system dx/dt = f(t,x(t), u(t)), tl < t < t2 constraints (tx(t))eA, uI(t) U(t, x(t)), toL,< t < t2, and boundary conditions *Research partially supported by NSF-grant GP-3920 at The University of Michigan. 1

(t, x(tl), t2, X(t2))CB, where A is a given subset of the tx-space E1 x En, where B is a given subset of the tlxlt2x2-space E2n+2, and where U(t,x) denotes a given closed variable subset of the u-space Ems depending on time t and space x. Here A may coincide with the whole space E1 x En, and U may be fixed and coincide with the whole space Em. In the particular situation where U(t,x) is compact for every (t,x), these problems reduce to Pontryagin problems; in the particular situation where U is fixed and coincides with the whole space Em, then these problems have essentially the same generality of usual Lagrange problems. Throughout this paper we shall assume U(t,x) to be any closed, subset of Em. In the present paper (II) we prove existence theorems for optimal (usual) solutions for the problem above. These contain as particular cases the Filippov existence theorem for problems of optimal control (U(t,x) compact), existence theorems for usual Lagrange problems (U = Em), and the Nagumo-Tonelli existence theorem for free problems (m = n, f = u). Also, we obtain as corollaries existence theorems for the case in which f is linear in u. In III we will discuss weak solutions as measurable probability distributions of usual solutions (Gamkrelidze's chattering states), and we will prove corresponding existence theorems in the present very general situation where U(t,x) is any arbitrary closed (not necessarily compact) variable set. All these existence theorems are proved by a new analysis of a minimizing sequence and by using the extensions of Filippov's clossure theorems proved in paper I, as a replacement for Toneili's semicontinuity argument. 2

To simplify references we continue the numeration of sections of paper I. References to paper I will be given by "I" followed by section number. In successive papers we shall extend the present results to multidimensional Lagrange problems involving partial differential equations in Sobole'v's spaces with unilateral constraints. 6. Notations for Lagrange Problems with Unilateral Constraints Let A be a closed set of the (t,x)-space E1 x En, and, for every (t,x)EA, let U(t,x) be a given subset of Em. Let fi(t,x,u), i = O,1,...,n, be continuous functions in the set M c E1 x En x Em of all (t,x,u) with u&U(t,x), (t,x)eA. Let f and f be the n-dim. and (n + 1)-dim. vector functions f= (fl, *f * n), f = fl,''',fn)' As usual we say that u(t) = (ul,,..um), x(t) = (xl...,xn), t < t < t2, is an admissible pair provided (a) u(t) is measurable in [tl,t2]; (b) x(t) is AC in [tl,t2]1 (C) [t,x(t) ]eA for every te[tl,t2]; (d) u(t) eU(t,x(t)), a.e. in [tl:,t2]; (e) fi(t,x(t), u(t)) is L-integrable in [tl,t2], i = O,l,..,n, and dxi/dt = fi(t,x(t),u(t)), i = l,.,.,n, a.e. in [tl,t2]. Thus, by introducing the auxiliary variable x, the differential equation dxo/dt = f,(t,x(t),u(t)), the boundary condition x~(tl) = 0, the vector x = (x~,x1,..,xn), and the set A = A x El c En+32) the pair [u(t),x(t)] is admissible if and only if the pair [u(t),x(t)] is admissible according to the definitions of no. 2 for the set A of the txspace E1 x En+l, the sets U(t,x) c Em, and the vector function f(t,x,u). If [u(t),x(t) ] is admissible, then u(t) is said to be an admissible control 3

function, x(t) a trajectory, and t2 x~(t2) = I[x,u] = fo(t,x(t),u(t))dt (1) tl the cost functional. A class n of admissible pairs x(t),u(t) is said to be complete if for every sequence xk(t), uk(t), tlk < t < t2k, k = 1,2,..., of admissible pairs all in I, with the sequence [xk(t)] converging in the metric p toward a vector function x(t) which is known to be a trajectory generated by some admissible control function u(t), then [x(t), u(t)] belongs to SQ. Such a class n is often defined in terms of boundary conditions. For instance, if B is a given closed set of points (tl,xl,t2,x2) of the (2n +2)-dim. Euclidean space E2n+2, we may define S as the class of all admissible pairs x(t),u(t) satisfying (tl,x(tl),t2,x(t2)) B. (2) Then S is a complete class in the sense mentioned above, since B is, by hypothesis, a closed set. We shall denote by B1 the projection of B on the (t,xl)-space En+l, that is, B1 is the set of all points (tl,xl)cEn+l for (tl,xl,t2,x2)eB. Analogously, we denote by B2 the projection of B on the (t2,x2)-space En+i. Obviously, B c B1 x B2, and B1 x B2 may be larger than B. It is often requested that each trajectory x(t) of a class Q as above possesses at least one point (t*,x(t*)) on a given compact subset P of A. Such a

condition is certainly satisfied if B is compact, or at least if B is closed and Bl, or B2, is compact. For the analysis of problems of Lagrange with unilateral constraints certain variable sets have to be taken -into consideration, namely, the set U(tx) above and the sets Q(t,x) = [zlz = f(t,x,u), ueU(t,x)] = f[tx,U(t,x)] c En, Q(t,x) = [zlz = f(t,x,u), ueU(t,x)] = f[t,x,U(t,x)] = [ = (z,)z = fo(t,x,u) z= f(t,x,u), ueU(t,x)] C En+l, Q(t,x) = [z = ( ) > f(t,xu), z = f(t,x,u), ueU(t,x)] C En+l The sets Q and Q are well known and have been considered by a number of authors (for instance, A. F. Filippov, [2]). The set Q(t,x) is being considered here and in [lc] for the first time. By considering this set, instead of Q or Q, we prove here theorems I and II (no. 7) which include a number of existence theorems for both problems of optimal control and the calculus of variations. 7. An Existence Theorem for Lagrange Problems with Unilateral Constraints Existence Theorem I. Let A be a compact subset of the tx-space El x En. and for every (t,x)eA let U(t,x) be a closed subset of the u-space E, Let f(t,xu) = (fo0,f...f n) = (fo,f) be a continuous vector function on the set M of all (t,x,u) with (t,x) A, ueU(t,x) Assume that, for every (t,x)eA the set 5

Q(t,x) = [= (z,z) Iz > fo(tx,u), z = f(t,x,u), uU(t,x)] c En+j is convex. Assume that U(t,x) satisfies property (U) in A, and Q(t,x) satisfies property (Q) in A. Assume that there is a continuous scalar function ~(), 0 < _ < + C, with $(t~)/5 + + +0 as 5 + 0O, such that fo(t,x,u) > ( lul) for all (t,x,u) eM, and that there are constants C,D > 0 such that jf(t,x,u) < C + DIuI for all t2 (t,x,u)eM. Then the cost functional I[x,u] =N fo(t,x,u)dt has an absolute minimum in any nonempty complete class Q of admissible pairs x:(t)-, u(t). If A is not compact, but closed and contained in a slab [to < t < T. xsEn], to, T finite, then theorem I still holds if, in addition, we know that (a) xLf +.... xnfn < F.txl2 + 1] for all (t,x,u)eM and some constant F > 0, and (b) every trajectory, in n contains at least one point (t*, x(t*)) on a given compact subset P of A (t* may depend on x(t)). If A is not compact, not contained in a slab as above, but A is closed, then theorem I still holds if hypotheses (a), (b) are satisfied, and (c) fo(t,x,u) >.t > 0 for all (t,x,u)eM with itl > R, for convenient constants p. > O, R > O. Finally condition (a) can be replaced in either case by the hypotheses: (a') There are constants G > 0 H > O such that fo(tx,u) > Glf(t,xu) I for all (t,x,u)eM with Ixl > H. Furthermore, when A is not compact but closed, both conditions fo >:(lul), Ifl < C + Diul can be replaced by the following set of conditions: (O)fo(txu)> - L for all (t,x,u)EM and some constant L > 0; j() fo(t,x,u) > p1 > 0 for all (t,x,u)eM with Itl > R, and some constants p. > O, R > O; (7) for every compact subset Ao of A there is a function 0 as above and constants Co, Do > O such that fo _> o ( lul), If _ Co + Dolui for all (t,x,u)EM with (t,x)eAo (where o, Co, Do may depend on Ao). 6

Proof of Existence Theorem I. We have T(~) > - Mo for some number Mo > O, hence i(S) + Mo > 0 for all > 0, and fo(t,x,u) + Mo > 0 for all (t,x,u)eM. Let D be the diameter of A. Then for every pair x(t), u(t), tl < t < t2, of 2 we have. t2 t2 I[xu] = fodt > - (ju)dt >- DMo > - oo Let i = Inf I[u], where Inf is taken for all u(t)Ef. Then i is finite. Let xk(t), uk(t), tlk < t < t2k, be a sequence of admissible pairs all in S1, such that I[xk, Uk] -* i as k + 00. We may assume n~~t2k~~i < I[Xk, Uk] = / fo(t, xk(t), uk(t))dt 0 be any given number, and let a = 2-1 (DMo + I i + 1). Let N > 0 be a number such that ~(z)/z > 1/a for z > No Let E be any measurable subset of [tlk,t2k] with meas E l/a, or uk < a( lukl), ao e in E2. Hence'uk(t) dt += (f + ) Uk(t) dt < N meas E1 +2 a ~( Iuk(t) j)dt 7

< N meas E + a [( uk(t) +M-]dt E2 < Nn + a 2 [( luk(t) I) +M0]dt!k t2k < NTI + tkI [fo(t,xk(t), uk(t)) +Mj]dt tlk < Nn + o(DMo + Iii + 1) < E/2 + e/2 = E. (2) This proves that the vector functions uk(t), tlk < t < t2k, k = 1,2,..., are equiabsolutely integrable. From here we deduce 4 xkt) at f(txk(t),uk(t) Idt < [A + Buk(t) | ]dt < A meas E + B luk(t) at, and this proves the equiabsolute continuity of the vector functions xk(t), tlk < t < t2k, k = 1,2,.... Now let us consider the sequence of AC scalar functions xo(t) defined by lk with

xk(tlk) =, Xk(t2k) = I[uk,xk] + i as k + + 00 i < xk(t2k) i + k 0 a.e. in [tlt2], and we define t t yk(t) =J u (t)dt, yk(t) = u(t)dt, tlk < t < t2k = 1, 2,o,o t'l k' tlk tSince -Mk < u(t) = O we have - tk o(t2k tlk) < Yk( ) _, Since -M < uk(t) = 0, we have' - Mo(t2k -tik) < yk(t) < 0, and the functions Yk(t) are monotone nonincreasing and uniformly Lipschitzian with constant Mo. On the other hand, the functions yk(t) are nonnegative, monotone nondecreasing, and uniformly bounded since O k 2k) k(t2k) + y (t2k)) - (t2k) (t2k) - Yk(t k) < i + 1 + M(t2k - tlk) < DMo + il + 1, By Aseoli's theorem we first extract a sequence for which yk(t) converges in the metric p toward a continuous function Y-(t), tl < t < t2, and Y-(t) is then monotone nonincreasing Lipschitzian with constant Mo, and Y-(tl) = O0 Then we apply Helly's theorem to the sequence Yk(t) and we perform a successive ex9

traction so that the corresponding sequence of the yk(t) converges for every t1 < t < t2 toward a function Yo(t), t1 < t < t2, which is nonnegative, monotone nondecreasing, but not necessarily continuous. This function Yo(t) is defined say at t1 only if oo-many of these intervals tlkt2k cover tlk, and so is for t2. If Yo(t) is not defined at t1 ot t2, we may define it at t by taking Y+(tl) 0, and at t2 by continuity at t2, because of its monotoneity. Thus 0 < Y+(t) < DMo + i + 1, t _ t t< t2. Finally, Yo(t) admits of a unique decomposition Yo(t) = Y+(t) + Z(t), tl < t < t2, with Y+(t1) = O, where both Y+(t),Z(t) are nonnegative monotone nondecreasing, where Y+(t) is AC, and Z'(t)= 0 a.e. in [tl.t2]. Finally, if Y(t) = Y-(t) + Y+(t), we see that xk(t), tl < t < t2, converges for all ti < t < t2, toward Y(t) + Z(t), where Y(t) is a (scaler) AC function, DMo < Y(t) < DMo + iil + 1, Y(tl) = 0. Let us prove that Y(t2) < i. For the subsequence [k] we o''` " it xk~t~k) - +k(t~k) C;Yk~t; have extracted last, we have t2k + t5, xk(t2k) + i, xk(t2k) = k(t2k) + Yk(t2k)If t2 is any point, tl t'2 < t2, t2 as close as we want 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, It2 - t2kl < 21t2 - t21I Then yYk(t2) - Y-(tak) <Mo t2 - t2kI <2Mc[2 t2| +' + +a f Since Yk(t) is nondecreasing, we have Yk.(t2) < Yk(t2k), and finally Yk(t2) + yk(t2) < Yk(t2) + yk(t2k) < Yk(t2k) +'yk(t2k) + lYk(t2) - Yk(t2k) 10

< xo(t2k) +2Molt2 - t21, where x~(t2k) + i as k + +oo, and XO(t2k) < i + k-'. Hence g- -1 Yk(t2 ) + Yk(t2) O. Thus Y(t2) =Y (t2) + Y (t2) < i +2Mlt2 - t21 As t2 + t2 - O, we obtain Y(t2) < i, since Y is continuous at t2. We will apply below closure theorem II to an auxiliary problem, we shall now define. Let u = (u~,u) = (uo,ul,~ 0,um), let UJ(t,x) be the set of all m 0~ ueEm+L with u = (u YU ) eU(t x), u > fo(t,x,u), let x = (x~,x) = (x~ xn), let f = f(t,x,u) = (fo,f) = (fo,fl,oo,fn) with fo = u~O Thus f depends only on t,xu (instead of tx 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 dx~/dt = u~(t), dx'/dt+ f (txu), i = oon, 11

with th'e constraints U(t) eU(t,x(t) ), or u~(t) > fo(t,x(t),u(t)), u(t) EU(t,x(t)), a.e. in [tl,t2], besides x~(tl) = 0, and [u,x]EQ. We have here the situation discussed in closure theorem II where x replaces x, x replaces y, x0 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 t2 t2 J[ux ] = fodu = u~(t)dt - x~(t2) -tl' tl Note that the set Q(t,x) = f(t,x,U(t,x)) of the new problem is the set of all z=(z0,z)6En+l such that z = u, since fo = u z = f(tox,u), u > fo(txu), uEU(t,x). Thus, the sets U, Q for this auxiliary problem are the sets U, Q considered at the beginning of this proof. We consider now the sequence of trajectories xk(t) = [xk~(t), xk(t) ], tlk < t < t2k, for the problem J[u,x] corresponding to the control function u(t) = [uk~(t), uk(t) ] with uk~(t) = fo(t,xk(t),uk(t)), uk(t)GU(t,xk(t)), and hence ~uk(t)U(t,xk(t)) tlk < t < t2k, k = 1,2,.... The sequence [xk(t)] converges in the metric p toward the AC vector function x(t), while xk~(t) + x~(t) as k + +oo for all te(tl,t2), and x~(t) = Y(t) + Z(t), where Y(t) is AC in [tl,t2] and Z'(t) = 0 a.e. in [tl,t2]. By closure theorem II we conclude that X(t) = [Y(t),x(t)] is a trajectory 12

for the problem. In other words, there is a control function u(t), tl < t < t2, (t) (u(t) = ( u (t)), with dY/dt = u~(t) > fo(t,x(t),u(t)), u(t) EU(t,x(t)), dx/dt = f(t,x(t),u(t)), (4) a.e. in [tl,t2], and t2 i > Y(t2) = J[xu] = u~(t)dt. (5) First of all [x(t),u(t) ] is admissible for the original problem and hence belongs to Q., since by hypothesis Q is complete. From this remark, and relations (4) and (5) we deduce t2 t2 i < I[x,u] =/ fo(t,x(t),u(t))dt < u~(t)dt < i, -tJ tl and hence all < signs can be replaced by = signs, u~(t) = fo(t,x(t),u(t)) a.e. in [tl,t2], and I[x,u] i. This proves that i is attained in 2. Existence theorem I is proved in the case A is compact. Let us assume now that A is not compact but closed, that A is contained in a slab [to < t < T, -co < xi < +o, i = l,...,n, to, T finite], and that the additional hypotheses (a) and (b) hold. If Z(t) denotes the scalar function Z(t) Ix(t) 12 + 1, then condition xlfl +.. xnfn < C(jx2 I + 1) implies Z' < CZ, and hence, by integration from t* to t, also 13

1 < Z(t) < Z(t*) exp CIt - t*| Since [t*,x(t*) ]eP where P is a compact subset of A, then there is a constant No such that Ix < No for every xeP, hence 1 < Z(t*) < No + 1, and 1 < Z(t) < (No2 + 1) exp C(T - to). Thus, for to < t < T, Z(t) remains bounded, and hence Ix(t) I < D for some constant D. We can now restrict ourselves to the consideration of the compact part Ao of all point (t,x) of A with to < t < T, IxI < D. Thus, theorem I is proved for A closed and contained in a slab as above, and under the additional hypotheses (a), (b). Let us assume that A is not compact, nor contained in any slab as above, but closed, and that hypotheses (a), (b), (c) hold. First, let us take an arbitrary element x(t), I(t) of Q and let j = I[x,i]. Then we consider an 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]. Now let 2 = t [Ijl + 1 + (b-a)Mo], and let [a',b' ] denote the interval [a -X, b + 2]. Then for any admissible pair (if any) x(t), u(t), tl < t < t2, of the class 2, whose interval [t1, t2] is not contained in [a', b'], there is at least one point t*e[tl, t2] with (t*, x*(t))eP, a<t* < b, and a point t-e[tl, t2] outside [a',b']. Hence [tl, t2] contains at least one subinterval, say E, outside [a,b], of measure > 2. Then I[x,u] > Bt - (b - a)Mo = jIl + 1 > i + 1. Obviously, we may disregard all pairs x(t), u(t), tl < t < t2, whose interval [tl, t2] is not contained in [a',b']. In other words, we can limit ourselves to the closed part A' of all (t,x)eA with a' K t K b'. We are now in the situation above, and theorem I is 14

proved for any closed set A under the additional hypotheses (a), (b), (c) Finally, we have to show that condition (a) can be replaced by condition: (a') There are numbers C, D > 0 such that fo(t,xu) > CIf(t,x,u)l for all (t,x,u)eM with Ixi > Do It is enough to prove theorem I under the hypotheses that A is closed and contained in a slab to < t < T, to, T as above, and hypotheses (a') and (b)o First let us take D so large that the projection P* of P on the xspace is completely in the interior of the solid sphere Ix < D, and also so large that D > T - to. Let u(t),x(t) be any arbitrary admissible pair contained in S, and let j denote the corresponding value of the cost functional. Let L = C [DMo + IJl + 1], and let us take Do = D + L. If any admissible pair u(t), x(t), tl < t < t2, of Q possesses a point (to, x(to)) with Ix(to) > Do, then x(t) possesses also a point (t*, x(t*))eP, with Ix(t*) I < D. Thus, there is at least a subarc r: x = x(t), x(t), t' < t < t", of x(t) along which Ix(t) I _ D and Ix(t) I passes from the value D to the value Do = D + L. Such an arc r has a length > L. If E = [t1, t2] - [t', t"], then rt2 r At rtit [xtuj J fdtdt >( ) fodt -DM + Cfdt - 0DM + C dx/dtldt = -DMo + CL > JjJ + 1 > i + 1' t As before we can restrict ourselves to the compact part Ao of all points (t,x) of A with to < t < T, Ixl < D. The case where A is closed, A is not contained in any slab as above, but conditions (a'), (b), (c) hold can be treated as before~ Theorem I is thereby completely proved. 15

Remark 1. If the set Q(t,x) = f[t,x,U(t,x) ] = [z = (z,) Iz = f(t,x,u), ueU(t,x) ] = [z = (Z,Z)Iz = fo(t,x,u), z =f(t,x,u), ueU(t,x)] I En+l is convex, then certainly the set Q(t,x) of theorem I is convex also. On the other hand, trivial examples show that Q(t,x) may be convex, when Q(t,x) is not. This is actually the usual case in free problems of the calculus of variations (see remark 2 below). Thus, the requirement in theorem I that Q(t,x) be convex for every (t,x) is a wide generalization of the analogous hypothesis concerning Q(tx) which is familiar in problems of optimal control. For these problems, Filippov's existence theorem is a particular case of theorem I. The theorem of A. F. Filippov [2]q As in theorem I, if A = El x En, if f(t,x,u) = (fof) = (fofly.a,*fn) is continuous in M, if U(t,x) is compact for every (t,x) in A, if U(t,x) is an upper semicontinuous function of (t,x) in A, if Q(tx) = f(txU(t,x)) is a convex subset of En+l for every (t,x) in A, if conditions (a) and (c) are satisfied, and the class Q of all admissible pairs for which x(tl) = xl, x(t2) = x2, tl,x1,x2 fixed, t2 undetermined, is not empty, then I[x,u] has absolute minimum in Q. This statement is a corollary of theorem I. Indeed, under hypothesis (c) we canrestrict A to the closed part A' of all (t,x)eA with a' < t < b' and Ix < N for some large N. If MO is the part of all (t,x,u) of M with (tx)EM, then the hypothesis that U(t,x) is compact and an uppersemicontinuous function of (tx) in Ao certainly implies that U(t,x) satisfies condition (U) in A0 and i 6

that Mo Is compact (I, no. 4, (vi) and (vii)). Also, since Q(t,x) is convex for every (tx) by hypothesis, we deduce that Q(t,x) is an uppersemicontinuous function of (t,x) and satisfies property (Q) (I, no. 4, (xiv) and (xiii)). Then Q(t,x) certainly satisfies property (Q) by the remark 1 above. Finally, since MO is compact, the growth condition fo > O and the remaining condition Ifl < C + Dlul are trivially satisfied. Thus, all conditions of theorem I are satisfied, and Filippov's theorem is proved to be a particular case of Theorem I. Remark 2. The analogous existence theorems of E. Roxin [8] and of L. Markus and E. B. Lee [9] are also essentially contained in Theorem I. For a detail on Roxin statement see Remark 4 below. Remark 3. For free problems of the calculus of variations (no. 11 below) we have m = n, U = En, f = u, hence Z(t,x) = f[t,x,U(t,x)] = [z = (z0,u) z0 =fo(t,x,u), uEn]c En+i Q(t,x) = [z (t,u) z >_fo(txu), ueEn]CEn+ the set Q is convex if and only if fo is linear in u, while Q is convex if and only if fo is convex in u. Thus condition Q convex of theorem I reduces to the requirement fo convex in u which is familiar for free problems in the calculus of variations. We shall prove in no. 11 that the Nagumo-Tonelli existence theorem for free problem is also a particular case of theorem I. Remark 4. The condition fo> ( Jul) with O(z)/z + +oo of the theorems above is said to be a growth condition on fo. As it is well known such a condition (for fo convex in u, and U=E) is equivalent to the condition that, for every (t,x)eA., we have 17

we have fo(t,x,u)/lul +oo as lul + +00 [L. Tonelli, 9a]. On the other hand, it is known already for free problems, that if such a condition is not satisfied at the points (t,x)eA-of even only one hyperplane t = t, then the absolute minimum need not exist. For free problems, bther additional conditions have been devised in such cases [9a]. Condition xlfl +. + + xnfn < C( Ix2 + 1) of theorem I can be replaced by x fl +..o + xnfn < cp(t)( Ix12 + 1), where @(t) > O0 is a fixed function of t which is L-integrable in any finite interval. The remark was made by E. Roxin [8] in analogous problems of optimal control. Remark 5. For problems of optimal control where U(t,x) is always compact we have given in [lbc] an existence theorem, say I*, similar to the Filippov's theorem above, where the condition "Q convex" is replaced by the following requirement: Q(t,x) is a convex subset of En, fo(t,x,u), ueU(t,x), is convex in u, and "the curvature of f is always small with respect to the convexity of fo" (see [lb], or [Ic] for a precise statement). Wherever this requirement implies the convexity of the set Q, then the theorem given in [lbc] becomes a corollary of Theorem I above. Also it should be pointed out that, whenever the relation -z = f(t,xu) between Q(t,x) and U(t,x) can be inverted and u = fi (t,x,z) is a continuous function of z in Q(t,x), then the set Q(t,x) can be represented by Q(t,x) = [z = (z~,z) z~ > F(t,x,z), zEQ(t,x) ], where F(t,x,z) = fo(tx, f-(t,x,z)), and thus the requirement of the convexity of the set: reduces to the requirement of the convexity of the function F(t,x,z) in z. Then the further requirement that Q(tx) satisfies property (Q) is certainly satisfied if, besides,F is quasi normally convex as proved in [I, no. 4, (xviii)]. We discussed in [Ilc] 18

a case where the requirement of theorem (I*) implies the convexity of F in u, and correspondingly (I*) becomes a corollary of I. The simpler requirement: Q(t,x) a convex subset of En and fo(t,xu) convex in u, does not suffice for existence, as we prove in the following number. 8. EXAMPLE OF A PROBLEM WITH NO ABSOLUTE MINIMUM The condition "Q(tx) convex for every (t,x)cA" of theorem I cannot be replaced by the simpler condition "Q(t,x) convex for every (tx)eA and fo(t,x,u) convex in u,"' not even when A and all sets U(tx) are compact (that is, for Pontryagin problems). This is shown by the following example. Let us consider the differential system: x = i(l - v) + [2 - 2(u - l)2]v y' = [2 - (u - 1)2 (1 - V) + uv 9 with t1 = 0, initial point (0,0), fixed target (0,1), and fixed control space [-1 < u < i 0 < v < o1. If zl = fl = u(l - v) + [2 - 2 (u- 1)2]v, Z2= f2 = [2 - 2V(u 1)2] (1 - v) + Ui w4e see that the segment [v = 1, -1 < u < 1] is mapped by f = (f1,f2) onto the arc of parabola ABC = [zl = 2 - 2-l(u - 1)2, z2 - u, -1 < u < 1], whose points A = (0,-I), B = (3/2,0), C = (2,1) correspond to u = -1,0,1 respectively. The segment Iv = 0,!- < u < 1] is mapped by f onto the arc DEF = [z1 = u, z2 = 2-2 19

(u - 1)2, -1 < u < 1], whose points -D = (-1,0), E = (0,3/2), F = (1,2) corres*pond to u = -1,0,1 respectively. Each segment [u = c, 0 < v < 1] is mapped by f onto the segment joining the points corresponding to (c,l) and (c,O) on the two parabolas. Thus, the image Q = f(U) of U is the convex body Q=(ABCFED) of the zl, z2-plane. Let us consider the cost functional t2 [x2 + (y - t)2 + v2]dt Far k 1,2,.2., let uk(t), vk(t), O < t < 1, be defined by taking uk(t) = -1, vk(t) -1 vk(t) = 0, or uk(t) = +1, vk(t) = 0, according as t belongs to the intervals k l(i - 1) < t < k-l(i - 1) + (2k), or k (i - 1) + (2k)1 < t < k-i, i = 1,20,,.,k. Then the functions xk(t), Yk(t), 0 < t < 1, satisfy the differential equations dx,/dt = +1, dyk/dt = 2, or dxk/dt = -1, dyk/dt = O0 according as t belongs to one or the other of the two sets of intervals above, Then xk(t) -> xo(t) = 0, Yk(t) + yo(t) = t uniformly in 0 < t < 1 as k + co. If Ck,Co denote these trajectories, we say that Ck + Co. The question as to whether Co is actually a trajectory, that is, whether there are admissible control functions uo(t), vo(t), 0 < t < 1, whose corresponding trajectory is Co can be answered in the affirmative because of the convexity of Qo Actually, the point (cOo,po)EU, aC% = 2 - 5 = -0.23607, po = (11)- 1(4 - A) = 0.16036, is mapped by f into (z1l = 0, z2 = 1), and thus uo(t) = C,, vo(t) = 50, 0 < t < 1, generate CO. Now we have xk(t)+ 0, Yk(t) + t, uniformly in [0,1] 20

(o2 + 2 + Bo2)dt-= o2 > O. Let us prove that I has no absolute minimum in the class Q of all trajectories satisfying the differential equations, boundary conditions, and constraints above. Indeed, I[Ck] + 0 shows that the infimum of I[C] in Q is zero, but this value cannot be attained by I in Q. Indeed, I[C] = 0 implies x = O, y = t, v = 0, and the first two relations alone imply u = %o, v = Po $ 0 a.e. in [0,1], a contradiction. Thus I cannot attain the value zero in Q. In this example Q is a convex set, fo is convex in (u,v), and even satisfies trivially the growth condition fo > p, since here U is a bounded set, Now let us prove that Q is not convex. It is enough to verify this for t = O, x = O, y = 0. Then Q is simply the set of all z = (zo, Zl,z2) with (Zl,Z2)EQ satisfying the relation zo > fo = v2, when zl, z2, u, v are related by z1 = f1, z2 = f2, (u,v)cU. Now the segment T = [V = O, -1 0 in Q-r, fo = 0 in r, and hence convex would imply that r is a segment, and this is not the case. This proves that Q is not a convex set. 9. ANOTHER EXISTENCE THEOREM FOR LAGRANGE PROBLEMS WITH UNILATERAL CONSTRAINTS Existence theorem II. Let A be a compact subset of the tx-space E1 x En, and, for every (tx)EA,'let U(t,x) be a closed subset of the u-space Em. Let f(tx,u) =fo f9,f-oDfn) = (fo,f) be a continuous vector function on the set M of all (txu) with (t,x) A, ueU(t,x). Assume that, for every (t,x) A, the set Q(tx) = i[z = (z~,z) ~E+llz~ > fo(t,x,u), z = f(t,x,u), ucU(t,x)] 21

is convex, and that U(t,x) satisfies property (U) and q(t,x) satisfies property (Q) in A. Let p(t) be a given function which is L-integrable in any finite interval such that fo(t,x,u) > cp(t) for all (t9x,u)eM. Let Q be a nonempty complete class of admissible pairs x(t), u(t) such that t2. Idx i-/dt I dt < Ni, i = 1,..,n, (24) tfor some constants N. > O, p > 1. Then the cost functional I[u,x] has an absolute minimum in Q. If A is not compact, but closed and contained. in a slab [to < T < T, -oo < xi < +o, i = 1,..,n, to, T finite], then theorem II still holds under the additional hypothesis (b) after theorem I. If A is not compact, nor contained in any slab as above, but A is closed, then theorem II still holds under the additional hypotheses (b) and (c*): fo(t,x,u) > cp(t) for all (t,xu)eM where +00 yp(t) is a given function which is L-integrable in any finite interval and. C p (t)dt = +co, p(t)dt = +oo. Finally, if for some i = l,..,,n, and any N > 0, -00 there is some Ni > 0 such that (x,u)cQ, I[x,u] < N implies Idxi/dtlPdt < Ni, then the corresponding requirement (24) can be disregarded. Proof of existence theorem IT. We suppose A compact, hence necessarily contained in a slab [to < t < T, to, T finite, -0 < xi < +oo, -i = l..o,n], and pt2 T then I[ux] =t fodt > P(t) Idt. This proves that the infinum i of to I[u,x] in 2 is necessarily finite. Let uk(t), xk(t), tlk < t < t2k, k = 1,2,.o., be a sequence of admissible pairs all in Q with I[uk,xk] + i. We may assume 22

t2k i < I[Xk, Uk] = fo(t,xk(t),uk(t))dt < i + l/k i + 1. (25) tlk Then t2k i Idxk/dtjPdt < Ni, i = l,...,n, k = 1,2,.... (26) tlk By the weak compactness of Ip we conclude that there is some AC vector function x(t) = (xl,...,xn), t <t < t2, such that tlk+ti, t2k +t2'', dxki/dt - dxi/dt weakly in Lp, xk(t) + x(t) in the p-metric. The proof is now exactly the same as for existence theorem I. If A is not compact, but closed and contained in a slab as above, and condi tion (b) holds, then for every admissible pair u(t), x(t) of Q we have Ix(t) - x(t*) I = t dx/dtdt dtl dt /P t 1/q tt < It t*I (N1 +...+Nn) where (t*,x(t*)) belongs to a fixed compace subset P of A. Then Ix(t*) I N', It - t*l < T - to, and Ix(t) I < N" for some constants N', N" > O. Thus, we can limit ourselves to the compact part Ao of all points (t,x) of A with to < t < T, I x < N". If A is not compact, nor contained in any slab as above, but A is closed and conditions (b), (c) hold, then we can use the same argument as for existence theorem I. Finally, we see that assumption (24) has been used only in (26) for a minimizing sequence ukxk. Since for a minimizing sequence we see already in (25) that I[uk,xk] < i + 1, it is obvious that any relation (24) which is a 23

consequence of a relation of the form I < M need not be required among the assumptions of theorem II. Theorem II is thereby proved. 10. EXAMPLES 1. Let us consider the (free) Problem t2 I[x] = ( + x' 2)dt = minimum, tl with x = (xl,.,,,xn), in the class Q of all absolutely continuous functions x(t) = (xl,...,xn), 0 < t < t2, whose graph (t,x(t)) joins the point (tl = 0, x(tl) = (0,...,0)) to a nonempty closed set B of the half-space t2 > O, xEEn. This problem can be written as a Lagrange problem: t2 J[x,u] = (1 + u(t) 2)dt = minimum dxi/dt= 1,,n ~dx/dtu', i l...n where x(t) = (xl,...,xn), u(t) = (u,un), m n, f = 1 +, i i = 1,...,n, and the control space U(t,x) is fixed and coincides with the whole space En. Here Q(t,x) = [(x,u) z > 1 + ul 2, ueEn] is a fixed and convex subset of En+1. The conditions of theorem III are satisfied with g = 1, go = 1, T(u) = O(IUI) = U12, or O(z) = z, 0 < z < +0, A is the half space A = [(t,x)l t > O, xeEn]cEn+l. Thus the problem above has an optimal solution.

2. The free problem I[x] = tx'2dt = minimum, x(O) = 1, x(1) = O 0 is known to have no optimal solution [6b]. The same problem can be written as a Lagrange problem with m = n = 1 in the form Jl[x,u] = tu2dt = minimum, x(O) = 1, x(l) = 0 0 dx/dt = u, u, El, as well as in the form J2[x,u] =J t3u2dt = minimum, x(O) = 1, x(l) = 0, 0 dx/dt = tu, uEEl The relative sets Q(t,x) are here subsets of the z z-plane E2. For the problem J1 2 the sets Q satisfy condition (Q), but fo = tu does not satisfy the growth condition of Theorem I. For the problem J2 the sets Q do not satisfy condition (Q). (We shall take into consideration the same sets under nos. 4 and 5 of Section 14 below). The same free problem with an additional constraint xt2 dt < No 25

where No > 1 is any constant, has an optimal solution by force of theorem II and subsequent remark. The optimal solution will depend on No. Note that No > 1 assures that the class Sn relative to the problem is not empty. Indeed for x(t) = 1 - t, we have f x'2dt = 1. 0 11. THE FREE PROBLEMS If we assume m = n, fi = ui, i = 1,...,n, U(t,x) = Ems then the differential system reduces to dx/dt = u, and the cost functional to t2.t2 I[u,x] = fo(t,x(t),u(t))dt = fo(t,s(t),x'(t))dt Then the problem under consideration in no. 6 reduces to a free problem (no differential system) where the integral is written in the form t2 I[x] = / fo(t,x(t),x'(t))dt, (1) and the only constraint is now (t,x(t)eA for all tl < t < t2. Again, complete classes a of vector functions x(t) can be defined by means of boundary conditions of the type (tl,x tl), t2, x(t2))~B,where B is a closed subset of E2n+2 as in no. 6. The Nagumo-Tonelli Theorem. If A is a compact subset of the tx-space E1 x En, if fo(t,x,u) is a continuous function on the set M = A x En, if for every (t,x)eA, fo(t,x,u) is convex as a function of u in En, if there is a continuous 26

scalar function cp( ), 0 < 5 < +oo, with 4(-)/ c + 0 as - ++00, such that fo *(t,x,u) > i( Jul) for all (t,x,u)eM, then the cost functional (1) has an absolute minimum in any nonempty complete class n of absolutely continuous vector functions x(t), tl < t < t2, for which fo(t,z(t),x'(t)) is L-integrable in [tl,t2]. If A is not compact, but closed and contained in a slab [to < t < T, xeEn], to, T finite, then the statement still holds under the additional hypotheses (T1) fo > CfuI for all (t,x,u)eM with IxI > D and convenient constants C > 0, D > 0; (T2) every trajectory x(t) of 2 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), (T2), and (T3) fo(tx,u) > p1 > 0 for all (t,x,u)eM with Itl > R. and convenient constants. > 0 and R > 0. Proof. First assume A to be compact. Then the set Q(t,x) reduces here to the set of all z = (z~,z) En+l with z~ > fo(t,x,z), z)En, where fo is convex in z, and satisfies the growth condition fo > cp( lul) with cp(5)/ - + + as + + +o. By the remark after lemma (xvi) of I, no. 4, fo is normally convex in u, hence quasi normally convex, and, by lemma (xvii), part (n), of I, no. 4, Qo satisfies condition (Q) in A. Thus, all hypotheses of theorem I of no. 7 are satisfied. If A is closed but contained in a slab as above then the condition (a) of theorem I reduces to u-x = C( Ix12 + 1) which cannot be satisfied since we have no bound on u. On the other hand, the condition (a') fo > C If for some C > 0 reduces here to requirement (T1) and condition (b) to requirement (T2). 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 (T3). All conditions of 27

theorem I are satisfied, and the cost functional (1) has an absolute minimum Q. 12. LAGRANGE PROBLEMS WITH f LITNEAR IN u. We shall consider now the case where all functions fi(t,x,u) i = l,...,n, are linear in u, and the control space U(t,x) is fixed and coincides with the total space Em. Precisely, we shall consider the Lagrange problem. t2 I[x,u] = [g(t,x) 0(u) + go(t,x) ]dt = minimum, (1) m dxi/dt =) gij(t,x)uj + gi(t,x), i = l,...,n, (2) j=1 where x = (xl,...,xn) Em, and D(u), uGEm, is a convex function of u satisfying a growth condition as in Nagumo-Tonelli theorem. If H(t,x) denotes the n x m matrix (gij(t,x)), and h(t,x) the n-vector (gi(t,x)), then the differential system (2) takes the form dx/dt = H(t,x)u + h(t,x) The sets Q(t,x), Q(t,x) relative to the problem above are Q(t,x) = [zlz = H(t,x)u + h(t,x), ucEm] c En (3) Q(t,x) [z = (z~,z) z~ > g(tx) (u) + go(t,x), z = H(t,x)u + h(t,x), ueEn] c En+l Obviously, Q(t,x) is a r-dimensional linear manifold in En where r is the rank of H(t,x). We shall need a few lemmas concerning the sets Q(t,x). 28

(i) If g is nonnegative, and 0 is nonnegative and convex, then both sets Q(t,x), Q(tx) defined in (3) are convex for every (t,x) eA. Proof. We give the proof for Q(t,x) Let = (g )0 = (qOy) be any two points of Q(tx), let 0 < a < 1, and z = (z~,z) = cx + (1 - -a)o. Then for some vectors ugveEm we have 0 > go(u) + go, ~ = Hu + h r~ 0>g(v) + g = Hv + h, z = az = o + (1 - a)no, z = ac + (1 - If weEm denote the vector w = cou + (1 - O)v, we have z = c + (1 -O) = o(Hu + h) + (1 - O)(Hv + h) H(au + (1 - c)v) + h = Hw + h, o= a~o + (1 - I)o > o(go(u) + go) + (1 - a)(go(v) + go) = g(ca(u) + (1 - a) c(v)) + g > > go(cmu + (1 - a)v) + go = go(w) + go Thus,9 z = (z~z) eQ(t x) and Q(t,x) is convex. (ii) If all functions 0, g, go, gij, gi are continuous, if T(u) is nonnegative asnd convex, and there is a function cp( ), 0 < 5 < +oo, such that cp( ) - +oo as - +, ald $(u) > cp( Iul) for all UeEm, if there is a neighborhood NW(t,x) of (t,x) where g > p1 for some constant p > O, then the set Q(t,x) defined in (3) 29

satisfies property (Q) at (t,x). Proof. We have to prove that Q(F,X) = nfcl co Q(t,x, b). It is enough to prove that n'cl co (t,x,) c Q(t,x) since the opposite inclusion is trivial. Let us assume that a given point z = ( -)~,z) nocl co Q(t,x,5) and let us prove that z = (z~,z)E(tx). For every 5 > 0 we have z = (z- ~)ecl co Q(t,x,6), and thus, for every 5 > 0, there are points z = (z~,z)c co Q(t,x,o) at a distance as small as we want from z = (z,z). Thus, there is a sequence of points k (zY,zk)e co Q(QT,x,zk) and a sequence of numbers 6k > O such that 6k -+, k z z. In other words, for every integer k, there are some pair (t,x0), (tk,xk), corresponding points z k = = (z kZk)cQk(tkxk), points ) CQk (tk I,, xi point s U~,UkEEm, and numbers ~k, 0 < C < 1, such that Zk = akzk + (1 - k)zk zk = CkZk + (1 - k)k Zk = akk + (1 - k)k Zk > g(t kx)c(U) + go(txk) Z = H(tkxI)ut + h(t1,x), Z" > g(ti,t xj) (uk) + go(t,xk) Z = H(t'k, x)ukj + h(t',xj), ( and suchthatt, X X, th t t, xj + x, Zk+ tZ x Z, Zk Z as k + oo. Obviously go(t,x) is bounded in N,(t,x), say go(t,x) > - G for G > O. The second relation (4) shows that of the two numbers zO' z0" one must be < Zk. It is not restrictive to assume that z' Zk for all k. Then the fourth relation (4) yields Zk > Zk' > g(tF,x~) (u ) + go(tg,xg ) 30

> G(uj) - G, where zk + z, and hence [zk] is a bounded sequence. This shows that 0(uk) < p1-( G +" Zk) hence [I(Uk)] is a bounded sequence, and finally [uk] is a bounded sequence because of the property of growth of 9. We can select a subsequence, say still [uk], which is convergent, say uk + u'EEm as k + o The sequence [ak] is also bounded, hence we can further select a subseqUence, say still [Ck]l for which [lCk] is also convergent. Thus uk -+ u', Ck + C. as k + oo. Let ukFEm be the point uk = Ckuk + (1 - Ck)uk. Then Zk akZj + (1 - -. zk = k[H(+tIx)u + h(t~,xl) ] + (1 - Cak) [H(tj,xj)uI + h(tI,xt) ] = H(tIXt) [QlkU1 + (1 - ak)Uk] + h(t",x4) + + OCkt[H(t2,xl) - H(t",xk) ]u + [h(tl,xk) - h(tk,xk) ]) (5) = H(t, xk). uk + h(tkl,x4) + Ak, zkO kZk + (1 - k) = = k[8g(ttx k)'(ut) + go(tt,xk) ] + (1 - Ck) [g(t9,xk)$(ui) + g o(t,xj) ] g(tkXky) [Cak$(u) + (1 - k)~(uk) ] + go(tkxk) + + (t - g(tkt9,x) g(t(u) + [go(tt,x) - go(tjPX,) 1] k> gQ4jf xk9 )k) + g( tk,xjk ) +k

Obviously Ak + 0, + 0, h(tk,x) h(t,x), go(tk,xj) + go(t,x). Since g(tj,xj) > 1i, we conclude as before that [I(uk) ] is a bounded sequence, and so is [uk], hence we can further select a convergent subsequence, say still [uk], with uk + u. Relations (5) yield now as k + 0o, z = H(t,x)u + h(t,x), o > g(t,x)~X(u) + go(t,X). Thus, z = ( z,z)Q(t,x), and statement (ii) is proved. Remark. Here are a few examples of linear problems and corresponding sets Q(t,x) and Q(t,x). 1. Take m 1, n. = 2, U = El, let ueEl be the control variable, and take D(u) = 1, g = 1, go = O, g11 = 1, gl2 = g2l = 0, g22 = t. Then the sets Q and Q depend on t, -1 < t < +1, and Q(t) = [z = ( zl 2) ~Z = Uz2, tU, -CO 1, 2 = tZ1, - < Z1 < + 00 C E Each set Q(t) is a straight line in E2 of slope t, and for each ~ > 0, the 2 set Q(0,5) contains both lines z = ~+zl, and the convex hul of Q(0,5) coincides with the whole plane E2. Thus Q(O) is the zl-axis and nSclco Q(O,5) is the whole zlz2-plane. The set Q(t) does not satisfy property (Q) at t = 0, and the same holds for Q(t). Here ~ = 1 does not satisfy the growth condition requested in (ii). 32

2. Take m = 1, n = 2, U = El, let uBE, be the control variable, and take O(u) = JuI, g = Itl, go = O, gll = 1, gl2 = g21 = 0, g22 = t. Then again the sets Q and Q depend on t only, gO = Itu =z2, Q(t) = [z = (1,z2) 1z2 = tz1, oo < z1 < +00] C E2 Q(t) = [ = (z zl,z2) |z~ > IZ21, t2 = tZ1, -c < t < +oo] E3 As before, the set Q(t) does not satisfy property (Q) at t = O. Analogously, for any ~ > 0, and -~ < t < ~, we see that z= (ZO,z11,z2) =, (1,5,l)EQ(), Zi= (ZOt?,z1f,z2 = (l, -1 1)( and, for a = 1/2, also'+ (1 - )z"= (z,zl,z2), _(Z z ( 1,0,1) E co Q(~,O) Hence, z - (1,O,l)enlc1co Q(O,co), z = (1,o,1))Q(O), and Q(t) does not satisfy property (Q) at t = O. Here g does not satisfy the condition g > t > 0 requested in (ii). 3. Take m = 1, n = 2, U = El, let ueEl be the control variable, and take I(u) = lul, g = 1, go = O, gll = 1, gl2 = g21 = 0, g22 = t. Then again the sets Q and Q depend on t only, and Qt=[= (zz2) I = tz, -_o < zl < +] C E2, 33

(t) = [ (zOz z 2)1z > Iz, z2 = t z -oo < z < +oo] c E As before, Q(t) does not satisfy property (Q), while'(t) does satisfy property (Q) at every t because of statement (ii)o 4. Take m = n = 1, U = El, let ueEl be the control variable, and take O(u) u2 g = t, go = O, g1l = t, G1 = O. Then Q(t) = [zlz = tu, -o tu2, z = tu, -oo < u < +oo] E2 How Q(O) is reduced to the single point z = O, while Q(t) for every t ~ 0 coincides with E1. Thus Q(t) does not satisfy roperty (Q) at t = 0. On the other hand Q(O) is the half straight line [z~ >0, z1 = o], while Q(t) for tK: 0 is the set Q(t) = z~ > t lz, -o0 < z < +]. Obviously,Q satisfies property (Q) at t = O (and at every t as well). 5. Take m = n = 1, U = El, let uEl1 be the control variable, and take O(u) = u 2 g = t s go = 0, gll = t. Then Q(t) = [zlz = tu, -o t32, = tu, -oo O, z1 = O] while Q(t) for t % 0 is the set Q(t) = [z0 > tz2, c < z < -o], and:cilco Q(O,) is the entire half plane [z~ > o, -oo < zl < +*]o Thus, neither Q nor Q satisfy property (Q) at t = O.

We shall denote by r(t,x) the:rank of the n x m matrix (gii(t,x)). Then O < r(tx) < min [m,n]. (iii) If all functions gij(t,x) are continuous, then r(t,x) < lim r(t,x) as (tx) + (t,x). The proof is a straightforward consequence of the continuity hypotheses. The statement below shows that a necessary condition for Q(t,x) to satisfy (Q) at (t,x) is that r(t,x) is constant in a neighborhood of (t,x), and this explains why the set Q of the examples 4 and -5 does not satisfy property(Q). On the other hand, the condition is not sufficient, as the sets Q of the examples 1i 2, 3 show since in these examples r 1 is constant. (iv) If all functions gij, gi are continuous in A, then a necessary condition in order that the set Q(t,x) satisfies condition (Q) at (t,x) is that r(t,x) = lim r(tx) as (t,x) +- (t,x) (thus, there is a neighborhood N (t,x) of (t,x) with r(t,xx)= r(t,x) for every (t,x)eN6(t,X). If Q(t,x) satisfies condition (Q) in A, then r(t,x) is a constant. Proof. Suppose that r(t,x) = r is not the limit of the (integral-valued) function r(t,x) as (t,x) + (t,x) Since r(t,x) = r < lim r(t,x) we must have r(t,x) = r < r + 1 < lim r(t,x). There is, therefore, a sequence (tk,xk), k 1,2,.oo,, with tk + t xk + x, and r + 1 < rk = r(tk,xk) < min [m,n]. The image of U = Em under the mappings H(tk,xk)u + h(tk,xk) and H(t,x)u + h(t,x) are, therefore, linear manifolds of En, say Q(tk,xk) of dimensions rk > r + 1, and Q(t,x) of dimension r. The images of u = 0 in Qk(tk, xk) and Q(t,x) are the points zk = h(tkxk), z = h(t,x). Let Bi~1 o be r orthonormal vectors in En such that 35

Q(tx) = [zeEEnlz = Z + ~lqr+...+~rrr, ~1.r real], and let us complete r1,..,rl into a system of n orthonormal vectors v1a,.e, qr' ~]r+1)ooo n. For every k, there are systems of rk orthonormal vectors llk, h' k of En such that Q(tkxk) = [zeE Iz = zk + ll k +...+ rk9rkk, -1*,S.rk real]. Since h(tkxk) +h(tx), H(tk,xk) h(tx), we can select o-k,~...kk so that, together with zk + z, we have also'jk i + O, 3 # i, j = l,...,rk as k + o. If we take 11 =... = ~r = Q r+l = 1, 5r+2 =... = Sn = O., then the point Zk = Zk + z+lk EQ(tkQxk). It is not restrictive to assume that for all k we have lZk - Zj <1/4, Ihjk'il < 1/4n, j #i, j =l, oork Then iTI~ = Xi=i (Br+ik' Ii)Ti ~.X) Ii = ir/, B i=r+r I' = ~ )i (qlr+1,k'qi)1i{ < 1i~f~$lk'qil < r(l/4n) < 1/4, IT"= I- r+l,k -1 > r+, H I > l1 - 1/4 3/4 Finally,

Zk - z = (Zk +r+,k) - r+k) <kl + zk - 1 + 1/2 = 3/2, and, for every zcQ(t,x), also - zi =!(Zk + lr+l,k) ( + jT111 +..+ rlr) n n i=l (Ir+1,k i)ri + i + (Zk - z) zr n ri=.l (-rrl+klir+lki i)i + (r+l i) i l - Iz >- Xjir+l (qr+l Qi)i -!zk - z = - Zk - i > 3/4 - 1/4 + 1/2. Thus, Iz - ZJ I 3/2, dist (zk, Q(t,x)) > 1/2 The sequence [zl ] is bounded, hence, it contains a convergent sequence, say still [zk] with zk + z'sEn, and Iz' - Zi < 3/4, dist (z', Q(t,x)) > 1/2 Finally, for every k there is a ukeU = Em such that = H(tk,xk)uk + h(tk,xk), or z7EQ(tk,xk), with Zk + Z'. Then z'ecl co Q(t,x,5) for every 5 > 0, and hence - ncn co z (tx,), z',Q(t,x)7 37

We have proved that Q(t,x) does not satisfy property (Q) at (t,x), a contradiction. This proves that, if Q(t,x) satisfies property (Q) at (t,x), then r(t,x)= lim r(t,x) as (t,x) + (t,x). The necessity of the condition is thereby proved. 13. EXISTENCE THEOREMSFOR LAGRANGE PROBLEMS WITH f LINEAR IN u. Existence Theorem III, Let us consider the Lagrange problem t2 I[x,u] - [g(t,x) 0(u) + go(t,x) ]dt = minimum, (1) m dxi/dt = =l gij(t,x)u + gi(t,x) i = 1,..,n, (2) where x = (x,... )En, u = (u,,um)Em, and 0(u) is a continuous nonnegative convex function of u. Assume that there is some continuous function cp( ), O <_ < +x, with cp()/~ + +oo as + +oo and c(u) > _p( lul) for every ueEm. Assume that all functions g(t,x), go(t,x), gij(t,x), gi(t,x) are continuous in A = E1 x En, and that g > 11 > 0, go > i> O I l giJ CI'g'" i j + gi+,g < Cg: 1i for some constants p. > 0, C > 0, and all (t,x)eA. Let n be the class of all pairs x(t), u(t), tl < t < t2, x(t) absolutely continuous, u(t) measurable, satisfying (2) aoe,, and such that the graph (t,x(t)) joins the fixed point (t, = 0, x(tl) = (0...o,O))eA to a given closed subset B of the half-space t 0, xcEn in Ao Then the Lagrange problem (1), (2) has an optimal solution in Qo 38

The functions N(u) = p( lul) = juIP, uEm, p > 1, as well as c(u) = cP( ul) = 0 for jul < C, c(U) = c(( |ul) = jujP - CP for lul > C, certainly satisfy the requirement for i0 Proof. By lemmas (i) and (ii) of no0 14 the set Q(tx) is convex for every (t,x) and satisfies condition (Q) in A. The set U = Em is fixed, closed, and obviously satisfies condition (U) Also fo(t,xu) = g(t,x)9 (u) + go(tx) and hence fo' liD(u) > 1ly( jul), fo > go >-kl, where t > 0, and hence both the growth condition for fo and the. conditions (c) and (d) of Theorem I are satisfied. Now if Ao is any compact subset of A = E1 x En then the continuous functions gij, gi are bounded in Ao, say Igij I < Co, Igi < Co, (where Co'depends on Ao), and If - IHu + hi < IHI IJ + Ihl < n2Co ul + c n for all (tx))Aoo Thus condition (5) of Theorem I is also satisfied. Condition (b) is satisfied since the initial point (tl,x(tl)) is fixed. Let us prove that condition (a7) is satisfied. Indeed cp( )/ + +c as ~ + +, hence p( )/~ > 1 for all i > D and some constant D > O. Then for Iuj > D we have IuI < cp( u ), and hence iul < D + cp( ju) for all u=Em' Now for all (tx)EA = E1 x E ar.d u FE we have ifj = Hu + hi IHI iul + Ihl < IHI(D + p( Jul)) + JhJ = HI n( jui) + (DIHI + |hl) < < Cg~(u) + (D + 1) Cgo = < c(D + l)(g (u) #+ go) = c(D + 1) fo 39

Thus fo > C-l(D + 1)-1 If for all (t,x,u)eEl x En x Em. All conditions of theorem I are satisfied, and the Lagrange problem I has an optimal solution. Existence Theorem IV. Let us consider the functional I[x,u] = [g(tx) (u) + go(t,x)]dt (1) with differential equations dxi/dt = gij(t,x)u + gi(t,x), (2) or dx/dt = H(t,x)u + h(t,x), where x = (x,...xn)"EEn u = (ul...,um)eU = Em, where H is the n x m matrix (gij) where h is the n-vector (gi) 9 and where ~(u) is a continuous nonnegative convex function of u. Assume that all functions g(t,x), go(t,x), gij(tx), gi(t,x) are continuous in A = E1 x En, and that g(t,x) > 0, go(t,x) > -GO for all (t,x)eA = E x En, go(tx) > 1 > 0 for all (tx)eA2E1 x En with Itl > Do, for some constants t > O, Go > O, Do > 0. Assume that the (convex) set (tx) = [ = (z~,z) |z0 > g~(u) + g, z = Hu + h, ucU = Em] c En+i satisfies condition i(Q) in A. Let 2 be the class of all pairs x(t), u(t), t1 < t < t2, x(t) absolutely continuous, u(t) measurable, satisfying (2) a.eo, and such that the graph (t,x(t)) joins the fixed point (tl = O, x(tl) =

(O,.o..,O))E1 x En to a given closed subset B of the half-space t > O, xeEn in E1 x En and such that I t2dxi/dtPdt -Ni, i = l,., n, (6) for some constants p > 1, Ni > O0 If Q is not empty then the Lagrange problem above has an optimal solution in a. The functions Oc(u) = O for lul, O(u) = ulP, p > 1, as well as (u) = 0 for Cul < c, P(u) = |u|P - cP for jul > c, p > 1, all satisfy the requirements above for T. The requirement go > i > 0 can be disregarded if B is contained in a slab [O < t < T, xEEn], T finite. Any requirement (6) which is consequence of t2 a relation.! (gT + go)dt < No can be disregarded. Proof. By (i) of no. 14 the set Q(t,x) is convex for every (t,x) in A. All conditions of Theorem II of no. 7 are satisfied, and thus IV is a corollary of II. Remark. The requirement concerning ~(t,x) of Theorem IV is certainly satisfied if we assume that (cz) g(tx) > ~i > 0 for all (tx)EA =El x En, (I) there exists a nonnegative convex function ep( ), 0 < < +mo with cp( ) +oo as + +oo and O(u) > cp( lul) for all usU < Em. Indeed, by statements (i) (ii) of no. 14 the convex set Q(t,x) satisfies property (Q) in A, Content, 11 The problem. I, p.3. - 2. The space of continuous vector 41

functions. I, p. 4 — 3. Admissible pairs x(t), u(t). I, p. 5. —4~ Uppersemicontinuity of variable sets. I, p. 6. —5. Closure theorems. I. p. 23. —6. Notations for Lagrange problems with unilateral constraints. II, p. 3. —7 An existence theorem for Lagrange problems with unilateral constraints. II, po 5.-o8. Example of a problem with no absolute minimum. II, p. 20. —9. Another existence theorem for Lagrange problems with unilateral constraints. II, p. 22. — 10 Examples. II, p. 25. —11. The free problems. II, p. 27. —12. Lagrange problems with f linear in u. II, p. 29. —13. Existence theorems for Lagrange problems with f linear in u.o II, p 39.

REFERENCES 1. L. Cesari, (a) Semicontinuita e convessita nel calcolo delle variazioni Annali Scuola Normale Sup. Pisa, 14, 1964, 389-423. (b) Un teorema di esistenza in problemi di controlli ottimi. Ibid., 1965, 35L78. (c) An existence theorem in problems of optimal control, J. SIAM Control, to appear. (d) Existence theorems in problems of Lagrange and optimal control, J. SIAM Control, to appear. (e) Existence Theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. I: Closure theorems. To appear. 2, A. F. Filippov, On certain questions in the theory of optimal control, Vestnik Mot~kow, Univ. Ser. Mat. Mech. Astr. 2, 1959, 25-32 (Russian). English translation in J. SIAM Control, (A) 1, 1962, 76-84. 30 R. Vo Gamkrelidze, On sliding optimal regimes. Dokl. Akad. Nauk SSSR 143, 1962, 1243-1245(Russian). English translation in Soviet Math. Doklady 3, 1962, 390-3954. L, Markus and E. Bo Lee, Optimal control for nonlinear processes, Arch. Rational Mech. Anal. 8, 1961, 36-58. 5. M. Naguno, Uber die gleichmassige Summierbarkeit und ihre Anwendung auf ein Variation Problem. Japanese Journ. Math. 6, 1929, 178-182o 6. L. So Pontryagin, Optimal control processes. Uspekhi Mat. Nauk 14, 1(85), 1959, 3-20. English translation in Automation Express 2, no, 1, 1959, 26-30. 7. L. SO Pontryagin, VO G. Boltyanskii, R. V. Gamkrelidze, and E. F, Mishchenko, The Mathematical Theory of Optimal Processes, Gosudarst, Moscow 1961. English translations: Interscience 1962; Pergamon Press 1964. 8. E. Roxin, The existence of optimal controls, Mech. Math, J. 9, 1962, 103119o 9. L. Tonelli, (a) Sugli integrali del calcolo delle variazioni in forma ordinaria, Annali Scuola Normale Sup. Pisa (2) 3, 1934, 401-450. Opere scelte, Cremonese, Roma 1962, 3, 192-254; (b) Un teorema di calcolo delle variazionli, Zanichelli, Bologna 1921-23, 2 vols. 10o Lo Turner, The direct method in the calculus of variations, A Ph.DI. thesis at Purdue University, Lafayette, Indiana, August 1957. 43

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