THE UNIVERSITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report No. 2 CLOSURE, LOWER CLOSURE, AND SEMICONTINUITY THEOREMS Lambe-rto Cesari! ORA:.Fr'j e.Pct 02 2416 submitted for: UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOSR-69-1662 ARLINGTON, VIRGINIA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1969

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TABLE OF CONTENTS Page 2.A. THE BASIC CLOSURE THEOREM 1 2.1. The Space of Continuous Vector Functions of a Scalar Variable 1 2.2. Orientor Fields 3 2.3. Meaning of Closure Theorems 4 2.4. Metric Upper Semicontinuity and Property (Q) 5 2.5. The Basic Closure Theorem for Functions of a Scalar Variable 8 2.6. Interpretation of Closure Theorem 1 in Terms of Usual Trajectories and Strategies. Closure and Compactness Theorems 12 2.7. Interpretation of Closure Theorem 1 in Terms of Generalized Solutions. Closure and Compactness Theorems 15 2.B. CLOSURE THEOREMS FOR FUNCTIONS WITH SINGULAR CONPONENTS 18 2.8. A Closure Theorem for Functions with Singular Components 18 2.C. LOWER CLOSURE THEOREMS 244 2.9. Lower Closure of Functionals in Integral Form 24 2.10. A Sufficient Condition for Lower Closure 26 2.11. A Variant of the Lower Closure Property 38 2.12. Criteria for Property (Q) of the Sets Q(t,x) 40 2.D. LOWER SEMICONTINUITY THEOREMS 50 2.13. Lower Semicontinuity of Functionals in Integral Form 50 2.14. Theorems of Lower Semicontinuity for Free Problems 52 2.15. Theorems of Lower Semicontinuity for Problems Depending on Higher Derivatives 55 iii

2.A. THE BASIC CLOSURE THEOREM 2.1. THE SPACE OF CONTINUOUS VECTOR FUNCTIONS OF A SCALAR VARIABLE We shall consider the collection X of all continuous n-dimensional vector functions defined on finite closed intervals of the real axis: x(t) = (x,...,x n), a < t < b, x(t) E E, where a,b are finite but not necessarily the same for all elements of the collection. It is often said that each element of X, or C:x = x(t), a < t < b, is a nonparametric continuous curve of the tx-space E1 x E. The orientation on C is then assumed to be the one corresponding to t increasing. The graph of an element x, or C, of X is the set of all points (t,x(t)) e E1 x En, a < t < b. For the sake of simplicity we shall denote by x and y, or x(t), a < t < b, and y(t), c < t < d, any two elements of X. We shall assign to X a distance function p(x,y) and by doing so we shall make X a metric space. To define p it is convenient to extend first x(t) outside the interval [a,b] by taking x(t) = x(a) for t < a and x(t) = b for t > b. The same we shall do for y(t). Then we take p(x,y) = la- cl + lb - dl + Maxjx(t) - t(t)l, where Max is taken for -a < t < +c. Because of the way in which x(t), y(t) are defined outside their original finite interval of definition, the maximum above exists. It is left as an exercise for the reader to prove the basic properties: (1) p(x,y) > 0, x,y E X; (2) p(x,y) = O if and only if a = c, b = d, x(t) y(t); (3) p(x,y) = p(y,x), x,y e X; (4) p(x,y) < p(x,z) + p(z,y), x,y,z e X. 1I

A sequence xk(t), ak < t < bk, k = 1,2,..., of elements of X is said to be convergent toward an element x(t), a < t < b, of X, provided p(xk,x) + 0 as k - c. Then ak * a, bk + b, and xk(t) + x(t) uniformly on (-w,+c) (after extension of each xk and x outside their original interval of definition as above). We shall say that the metric p defines the uniform topology on X. As mentioned earlier we shall consider arbitrary classes S of elements of X, or continuous curves in the tx-space El x E. For our purpose the following simple statement shall be noted. We shall assume that A is any closed subset of the tx-space E1+, that B is a closed subset of the tlxlt2x2-space E2n+2, and that g(tl,xl,t2,x2) is a continuous scalar function on B. We shall denote by 2 the class of continuous vector functions satisfying (t,x(t)) e A for all tl < t < t2, and (tl,x(tl),t,xt2,x(t2)) E B. Thus C X. (2.1.i). The class Q is closed in the uniform topology (metric p) and the functional I[x] = g(tl,x(tl),t2,x(t2)) is continuous in Q. In other words, if xk(t), tlk < t < t2k, are elements of X with (t,xk(t)) E A, (tlk, xk(tlk),t2k,xk(t2k)) E B. and x(t),tl < t < t2, is an element of X such that p(xk,x) x 0 as k + o, then (t,x(t)) E A, (tl,x(tl),t2,x(t2)) E B, and I[xk] + I[x] as k oo. Proof of (2.1.i). First p(xk,x) + 0 implies tlk + t, t2k > t2 as k o. Secondly, x(t) is continous in [tlt2], hence its extension is continuous in (-oo,+co); hence, given ~ > O0 there is 6 > 0 such that It - tll < t implies Ix(t) - x(tl)l < E, and It - t2j < 8 implies Ix(t) - x(t2)| < e. Finally, there is k such that k > k implies Itlk - tll < S, It2k - t2J <, P(Xkx) <, and hence 2

Xk(tlk ) - x(tl) I < Jxk(tlk) - X(tlk)) + IX(tlk) - x(tl)J < 2~, and analogously Ixk(t2k) - x(t2) < 2g for k > k. This proves that xk(tlk) + x(tj), xk(t2k) * x(t2) as k > o. Thus, (tlk,Xk(tlk)t2k (t2)) B with B closed implies (tl,x(tl),t2,x(t2)) E B. By the continuity of g then g(tlk, xk(tlk)t2k, Xk(t2k)) * g(tl,x(tl),t2,x(t2)) as k oo. Also, (tlk,Xk(tlk)) C A with A closed implies (tl,x(tl))E A, and analogously we prove that (t2,x(t2)) e A. Finally, for every t c (tl,t2), we have also t E (tlk,t2k) for all k sufficiently large, and (t,xk(t)) e A then implies (t,x(t)) E A. We have proved that (t,x(t)) E A for all tl < t < t2, hence x is an element of Q, and this completes the proof. Finally, we remind here that Ascoli's theorem in terms of the metric p holds in the following form: If xk(t), tlk < t < t2k, k = 1,2,..., are equicontinuous vector functions, and there is a constant M such that -M < tlk < t2k < M, Ixk(t) I M for t E [tlk,t2k], k = 1,2,... then there is a subsequence [xk ] convergent in the metric p toward a contir:aous vector function s x(t), tl < t < t2. 2.2. ORIENTOR FIELDS We assume that a set A is given in the tx-space E1 x E, x = (x1,...,x )1 and that for every (t,x) c A a set of "allowable directions" z is assigned, precisely, a nonempty set Q(t,x) of vectors z = (z l,,z ) is given, Q(t,x) {Ztxzt and this set may depend on (t,x) E A. As mentioned in 1.2 we denote the relation dx/dt E Q(t,x) (2.2.1) 35

an orientor field. A solution x(t), t1 < t < t2, of (2.2.1) is a vector-valued function x(t) = (xl,...,xn), tl < t < t2, such that (1) x(t) is absolutely continuous (AC) in [tl,t2]; (2) (t,x(t)) E A for all t C [tl,t2]; (3) dx/dt e Q(t,x(t)) almost everywhere (a.e.) in [tl,t2]. Thus, for almost all t E [tl,t2] the direction dx/dt = (xl,...,x ) of the curve x = x(t) at (t,x(t)) is one of the "allowable directions" z e Q(t,x(t)) assigned at (t,x(t)). An orientor field will be said to be autonomous if Q(t,x) depends on x only and not on t. Nevertheless, every orientor field can be written as an autonomous one by a change of coordinates. Indeed, if we add the vector variable x satisfying the differential equation dxo/dt = 1 and initial condition x (tl) tl, and then we use the (n + 1)-vector x = (x, x,...,x ), and direction set ~_ ~ 0 n 0 o Q(x) = [z = (z Z =(z,z), z e Q(x,x), z = 1], then system (2.2.1) becomes dx/dt e Q(x). We may use this remark in proofs in order to simplify notations. Remark 1. For the sake of simplicity, we have assumed the variable x to vary in a Euclidean space E. As a careful reader may see, most of the results ben low are valid even if we allow x to vary in much more general spaces, and attention will be called to this fact when needed. 2.3. MEANING OF CLOSURE THEOREMS The closure theorems we shall discuss below answer affirmatively a very important question, a question which is relevant even for the particular case of

ordinary differential systems. If we have a sequence [Xk] of solutions xk(t), tlk < t < t2k, of an orientor field (2.2.1), and [xk] "converges" toward a given function x (t), t1 < t < t2, then also x is a solution of the orientor field (2.2.1). This is essentially true, though under various sets of assumptions on A, on Q, on the mode of convergence, on x In the closure theorems below we shall always assume that A is a given closed subset of the tx-space E1 x E, and that, for every (t,x) e A, Q(t,x) is a closed convex subset of the z-space E. Concerning the mode in which Q(t,x) is allowed to vary as (t,x) describes A, we shall need a very mild property which is usually described as an upper semicontinuity property. Such a property has been introduced by various authors in different ways for different purposes, and we shall introduce some of these definitions as we go along and we need them. 2.4. METRIC UPPER SEMICONTINUITY AND PROPERTY (Q) Given any set Z in E, we shall denote by cl Z, bd Z, co Z the closure of Z, the boundary of Z, and the convex hull of Z, respectively. Thus, cl co Z denotes the closure of the convex hull of Z, or briefly the closed convex hull of Z. Given a set A of the tx-space E1 x E, a point (t o,X) e A, and a number 6 > 0, we shall denote by N6(t,xo), or neighborhood of (to,Xo) in A, the set of all (t,x) E A at a distance < 6 from (to, Xo). Then N (to,x) c A. Also, given — 6 o a set U(t,x) of points z = (zl,...,z ) for each (t,x) e A, a point (t o, ) C A, and a number 6 > 0, we shall denote by U(to,x;6) the union of all U(t,x) with (t,x) E N (to,xo), in other words

U( t, x0, 6 ) U U( t, x) (t, x) cN( to, Xo) Thus U(t,xo) c U(to,x,) for all i > 0. We say that U(t,x) is metrically upper semicontinuous at a point (t,x ) E A provided, given ~ > 0, there is some 5 = F(t,x,~) > 0 such that U(t,x) C [U(to, Xo)] for all (t,x) E N,(to,x0) and where [U]~ denotes the closed Eneighborhood of Q. We say that U(t,x) is metrically upper semicontinuous in A if it has this property at every (t,x) E A. The following analogous property of upper semicontinuity is of interest. We shall say that a variable set U(t,x), (t,x) E A C E1+, U(t,x) c E, has property (U) at a point (to,Xo) E A provided U(t,Xo) = As cl u (to, X;)'0' 0 b>0 0 that is, U(t X) = n cl I U U(t,x)) 0 0 6>0 (tX)EN E (t0 We shall say that U(t,x) has property (U) in A if it has property (U) at every point (to,Xo) e A. If M denotes the set of all (t,x,u) with (t,x) E A, u E U(t,x), then M C E1+n+m, and M is the graph of U(t,x). The main statement concerning property (U) is the following one: (2.4.i). If A is closed, then M is closed if and only if U(t,x) has property (U) in A. We shall prove this statement in [App. A.l.ii] together with others. If U(t,x) has property (U) then U(t,x) is certainly closed as the intersection of 6

closed sets. Property (U) is usually denoted as Kuratowsky's upper semicontinuity. In the proof of the Closure Theorems below, we shall often require that the sets U(t,x) be closed, convex, and satisfy an analogous property of upper semicontinuity, which we shall denote as property (Q), and which is specific for closed and convex sets. We shall say that U(t,x), (t,x) E A, has property (Q) at the point (to,xo) E A provided U(t,xo) = N cl co U(to,xO,), (2.4.1) o' 0 >0 Xo that is, U(t x) = N cl co { U(tx) o 0 6>0 (t,x)eN6(toX) U(tx) We shall say that U(t,x), (t,x) E A, has propertyQ) in A if it has property (Q) at every (t,x) e A. If U(t,x) has property (Q) then certainly U(t,x) is closed and convex as the intersection of such sets. For closed convex sets, property (Q) is a way to express the idea of upper semicontinuity in a form which is more general than the metric upper semicontinuity. The concept of metric upper semicontinuity can be traced very far back (see for instance, F. Hausdorff [ 1]). Note that we can always denote (t,x) as a unique variable, say x E En+l' hence A is a subset of the x-space En+1l and U(x) depends on x only. We shall use this remark in the proofs for the sake of simplifying notations. (2.4.i). If A is closed, if, for every (t,x) E A, U(t,x) is a closed subset of Em, and U(t,x) is metrically upper semicontinuous in A, then certainly 7

U(t,x) has property (U) in A. If all sets U(t,x) are closed and contained in a given interval in E, then U(t,x) has property (U) in A if and only if U(t,x) is metrically upper semicontinuous in A. If, for every (t,x) E A, the sets U(t,x) are known to be closed and convex, then the same statements above hold for property (Q). The proof is given in App. A (A.l.v; A.l.vii; A.2.iv; A.2.v). Remark 1. Property (U) for closed sets, and property (Q) for closed convex sets are actually more general than metric upper semicontinuity. This can be shown by the following simple example with m = 2, A = [tlO < t < 1], U(t) = [z = (Z1,z2)10 < Zl < + 0, 0 < Z2 < tzl]. Here U(t) is an angle, and obviously, for t > t, U(t) is not contained in any [U(t )], no matter how t is close to t O o U(t) has both properties (U) and (Q) in A, but it is not metrically upper semicontinuous. More on the comparison properties of upper semicontinuity will be given in App. A. 2.5. THE BASIC CLOSURE THEOREM FOR FUNCTIONS OF A SCALAR VARIABLE (2.5.i). Closure Theorem 1. Let A be any closed set of the tx-space El x E, for every (t,x) e A let Q(t,x) be a closed convex subset of points z = (zl,...,z ), and let us assume that Q(t,x) satisfies property (Q) at every point (t,x) E A with exception perhaps of a set of points whose t coordinate lies on a set H of measure zero on the t-axis. If xk(t), tlk < t < t2k, k = 1,2,..., is a sequence of solutions of the orientor field (2.2.1), convergent in the p-metric toward an AC function x(t), tl < t < t2, then x(t) is also a solution of the orientor field. In other words, we know that each xk(t), tlk < t < t2k, k = 1,2,..., is AC, 8

that (t, xk(t)) E A for every t E [tlk, t2k], and that dxk/dt E Q( t, xk(t)) a.e. in [tlk, t2k], we know that p(xk,x) * O, hence tlk + tl, t2k + t2, as k + o, and that x(t) is AC in [tl,t2], and we want to prove that (t,x(t)) E A for all t c [tl,t2], and that dx/dt e Q(t,x(t)) a.e. in [tl,t2]. Proof of (2.5.i). The vector functions x'(t), tl < t < t2, x'(t), t < t < t k 1k - 2k' are defined a.e. in [tl,t2] and [tlk,t2k] respectively, k = 1,2,..., and are L-integrable in the respective intervals (that is, each component is L-integrable). Since p(xk,x)'- 0, hence tlk + tl, t2k + t2 as k -*, if t E (tl,t2), or tl < t < t2, then tlk < t < t2k for all k sufficiently large, and (t,xk(t)) E A. Since xk(t) + x(t) as k - m and A is closed, we conclude that (t,x(t)) E A for every tl < t < t2. Since x(t) is continuous in [tl,t2], 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 [tl,t2] the derivative x'(t) exists and is finite and t E [tlt2] - H. Let t be such a point with tl < t < t2. Then, there is a 0 0 a > O with tl < to - a < to + a < t2, and, for some k and all k > k, also tlk < t a < t + a < tk Let x = x(t). We have xk(t) + x(t) uniformly in [t - a,t + a] and all functions x(t), xk(t) are continuous in the same interval. Thus, they are equicontinuous in [t - a,t + a]. Given E > 0, there is 3 > 0 such that t,t' E [t - a,t + a], It - t' I <, k > k, implies 0 0 - 0 Ix(t) - x(t')l < 6/2, Ixk(t) - xk(t') _ < e/2. (2.5.1) We can assume 0 < 5 < a, 68 ~. For any h, 0 < h < 6, let us consider the averages 9

mh = h-1 f x'(t + s)ds = h- l[x(t + h) - x(t )], (2.5.2) mhk = h- foh Xk(t + s)ds = h-[xk(t + h) - x(t (2.53) Given rj > O0 we can take h so small that Imh - x'(t )J < (2.5.4) Having so fixed h, let us take k1 > k so large that lmhk- mhl < r, Ixk(to) - x(to)J < e/2 (2.5.5) for all k > k1. This is possible since xk(t) - x(t) as k + oo both at t = t and t = t + h. Finally, for 0 < s < h, Ixk(t + s) - x(t ) l < Ixk (t + s)- xk(t )j + xk (to)- x(t )I < /2 + /2:= I(to + s) - to = s < h < 6 < ~, xk(t + s) e Q(t + s,xk(t + s)) a.e. Hence, for almost all s, 0 < s < h, xk(to + s) Q(t,x 2) and consequently x'(t + s) E cl co Q(t,xo, 2), a.e. in [O,h]. k 0 00 The average mhk as defined by (2,5.3) is then also a point of the same closed and convex set, or mhk E cl co Q(t,x,2~) 10

for the chosen h and every k > kl. By relations (2.5.4) and (2.5.5) we deduce Ix'(to) - mhkl < Ix'(to) - hl + Imh - mhk < 2r, and hence x'(t) E [cl co Q(to, Xo,2E) ]2 Here 5 is an arbitrary number, and the set in brackets is closed, hence x'(t ) E n [cl CO Q(to,x,2]2 = cl co Q(to, Xo,2), for every ~ > O. Thus, by property (Q), x'(t ) E nf cl co Q(to,x, 2~) = Q(t X ). We have proved that for almost all t e [tl,t2], we have dx/dt E Q(t, x(t)) The Closure Theorem 1 is thereby proved. The following example illustrates the first closure theorem. Let n = 1, A = E2, Q = Q(t,x) = [z I-l < z < 1], and xk(t), 0 < t < 1, k = 1,2,..., be defined by xk(t) = t - ik-l if ik-1 < t < ik-l + (2k)-1, xk(t) = (i + l)k'1 - t if ik- + (2k)-1 < t < (i + l)k-1 for i = O,l,...,k - 1. Then xk(t) - x (t) = 0 uniformly in [0,1]. On the other hand, x'(t) = +1 according as t is an interior point of one or the other of the two sets of intervals above, x'(t) = 0, and 0 xk(t), x'(t) e Q for almost all t. Here Q is a closed convex set. If we had taken Q = Q(t,x) = [zlz = -1 and z = +1], then obviously xk(t) E Q while x'(t) ~ Q. Here Q is closed but not convex. 11

2.6. INTERPRETATION OF CLOSURE THEOREM 1 IN TERMS OF USUAL TRAJECTORIES AND STRATEGIES. CLOSURE AND COMPACTNESS THEOREMS (a) Let us assume here that A is any closed set of the tx-space E1+n, x = (xl,...,x ), that, for every (t,x) E A, U(t,x) is a subset of the u-space E, u = (ul,...,u ), that the set M of all (t,x,u) with (t,x) E A, u c U(t,x), is closed, that the vector function f(t,x,u) = (fl,...,f ) is continuous on M, that B is a closed subset of the tlxlt2x2-space E 2n+2, and that g(tl,xl,t2,x2) is a continuous scalar function on B. Also we assume that the sets Q(t,x) = f(t,x, U(t,x)) c En are closed, convex, and satisfy condition (Q) at every point (t,x) E A-with exception perhaps of a set of points whose coordinate t lies on a set of measure zero on the t-axis. As stated in (1.1), we say that a pair x(t(t), t), tl < t < t2, is admissible (and that x is a (real) admissible trajectory provided x is AC in [tl,t2], u is measurable in [tl,t2], (t,x(t)) A for all t c [tl,t2], and u(t) E U(t,x(t)), x' = f(t,x(t),u(t)) a.e. in [tl,t2]. For x,u admissible we define as cost' fuinctionalI[x,u] = g(tl,x(tl),t2,x(t2)). In other words, we have a Mayer problem with state variables x = (xl,...,x), m and control variables u = (ul,...,u ) (2.6.i) (a closure theorem). Under the hypotheses above (in particular the sets Q(t,x) being convex), any AC limit x(t), tl < t < t2, in the p-metric for trajectories is a trajectory. In other words, if xk(t), uk(t), tlk < t < t2k, k = 1,2,..., is any sequence of admissible pairs, if x(t), tl < t < t2, is any AC function, and p(xk,x) + 0 as k + o, then x is a trajectory, that is, there is a measurable function u(t), tl < t < t2, such that x(t), u(t), tl < t < t2, is an admissible pair, and I[xk,uk] + I[x,u] as k + c. Note that, whenever we wish to disregard boundary conditions, we have only to take B=E2 +2 and g and I need not bedefined. 12

Proof. By (2.l.i), (t,x(t)) e A for t e [tl,t2], and (t1,x(tl),t2,x(t2)) e B. Also, x'(t) = f(t,xk(t),uk(t)) e Q(t,xk(t)) for almost all t E [tlk,t2k], i = 1,2,.... By (2.5.i), or closure theorem 1, then x'(t) E Q(t,x(t)) a.e. in [tl,t2], and by (1.6.i), or implicit function theorem for orientor fields, there is a measurable function u(t), tl < t < t2, with u(t) E U(t,x(t)), x'(t) = f(t,x(t),u(t)) a.e. in [tl,t2], that is, x,u is an admissible pair. By (2.1.i) we know that I[xk,uk] + I[x,u]. For Lagrange problems with usual functional I = ft2 f (t,x(t),u(t))dt tjwith f (t,x,u) a continuous scalar function on M, the class of admissible pairs is restricted to only those for which f (t,x(t),u(t)) is L-integrable on [tl, t2]. By introducing the variable x, the additional differential equation dx /dt = f (t,x(t),u(t)), and condition x (tl) = 0, we have I = x (t2), and we have again a Mayer problem in the state variable x = (x,x) = (x,x,...,x ) and control variable u = (ul,...,u ). Closure statement (2.6.i) could now be repeated in the new situation, with xk,x replacing xk,x, by assuming that the sets Q(t,x) = f(t,x,U(t,x)) c E +1 are closed convex, and satisfy property (Q) f = (f,f) = (f ofl, ".,fn), and by demanding that p(xk,x) - 0 as k -* o. This last requirement is in most cases too demanding in Lagrange problems with unbounded controls, and we shall introduce and discuss the concept of lower closure in (2.9). (b) Instead of the hypotheses stated at the beginning of (a), let us now assume that A is compact, that M is compact, that f(t,x,u) is continuous on M, that B is closed, and that g is continuous on B. Also, let us assume that the sets Q(t,x) = f(t,x,U(t,x)) are convex for all (t,x) E A, with exception perhaps 13

of a set of points whose t coordinate lies on a set of measure zero on the taxis. We have as before a Mayer problem with cost functional I[x,u] = g( t1n X( t1)1 t22 X( t2))(2.6.ii) (a compactness theorem). Under the hypotheses above (in particular, the sets Q(t,x) being convex), the family of all usual trajectories x(t), tl < t < t2, is sequentially compact in the p-metric. In other words, if xk(t), uk(t), tlk < t < t2k, k = 1,2,..., is a sequence of admissible pairs, then there is a subsequence [kS], and an AC function x(t), tl < t < t2, such that p(xk,u) -* 0 as k + mo, and x is a trajectory, that is, there is a measurable function u(t), tl < t < t2, such that x(t), u(t), t1 < t < t2, is admissible, and I[xk, k] - I[x,u] as k -+ a. Proof. Here A is compact, hence bounded, and A is contained in some set ItI < L, IxI < L. Also, f is continuous on M, hence bounded, and we can take L in such a way that Ifl < L on M. Then -L < t lk< t2k < L for all k, and Ixk(t) l < L for all t E [tlkt2k] and all k. Also, xk is AC and Ixk(t) = If(t,xk(t),uk(t)l < L a.e. in [tlkt2k], hence Ixk(t) - xk(t')I <Lit - t'l for all t,t' C [tlk,t2k] and all k. Thus, the trajectories xk are equibounded and equilipschitzian, hence equicontinuous. By Ascoli's theorem (see (2.1)) there is a subsequence [ks ] and a continuous vector function x(t), tl < t < t2, with P(Xk,x) + 0 as k + a. Then x is Lipschitzian of the same constant L, hence AC in [tt2s]. Now the set M is compact by hypothesis, hence by (A.l.v) the compact sets U(t,x) are metrically upper semicontinuous in A, and by (A.3.i), the sets Q(t,x) are also compact and metrically upper semicontinuous in A. From (A.2.v) we conclude then that the sets Q(tx) which are convex by hypothesis,

have property (Q) in A. Statement (2.6.ii) follows now from (2.6.i). We may now consider Lagrange problems as in (a). Again we shall assume A and M compact, and we take a scalar f (t,x,u) continuous on M. Note that now for x(t), u(t), tl < t < t2, admissible in the usual sense, the function fo(t,x(t),u(t)) is necessarily measurable and bounded, hence L-integrable. Actually, the functions f (t,x(t),u(t)) are equibounded, and so are the functions f(t,x(t),u(t)) = (f,fl,...,f ). Statement (2.6.ii) holds now also in the new situation, provided we assume that the sets Q(t,x) = f(t,x,U(t,x)) are convex, and the conclusion is that the trajectories x(t), tl < t < t2, are now sequentially compact, with x = (x,x) = (x,x1,...,x ), and dxo/dt = f (t,x(t), u(t)), x0(tl) = O. (See (2.9),(2.10) for extensions of these statements ) 2.7. INTERPRETATION OF CLOSURE THEOREM 1 IN TERMS OF GENERALIZED SOLUTIONS CLOSURE AND COMPACTNESS THEOREMS Here we completely abandon all hypotheses of convexity of the sets Q and Q of (2.6). (a) Let us assume here that A is any closed set of the tx-space E1 n+n' x = (xl,...,x ), that for every (t,x) e A, U(t,x) is a subset of the u-space Em, u = (u1,..., u), that the set N of all (t,x,p,v) E E with (t,x) E A, l+n+m+my = (P P) e r, V = (U),...,u()) E V(t,x) = [U(t,x)]7 is closed, that the vector function f(t,x,u) is continuous on the set M = [t,x,u) (t,x) e A, u E U(t,x)], that B is a closed subset of the tlxlt2x2-space E2n+2, and that g(tl,xl,t2,x2) is a continuous function scalar on B. Also we assume that the sets R(t,x) = co Q(t,x) = co f(t,x,U(t,x)) C En are closed, convex, and satisfy property (Q) at all points (t,p E A with exception perhaps of a set of points (t,x) whose t coordinate lies on a set of measure zero on the t-axis. As stated in 15

(1.8), we say that x(t), p(t(t), t), tl < t < t2, is an admissible generalized solution (and that x is a generalized trajectory), provided x is AC in [tl,t2], p(t), v(t) are measurable in [tl,t2], (t,x(t)) c A for all t E [tl,t2], p(t) C r, v(t) e V(t,x(t)), x'(t) = h(t,x(t),p(t),v(t)) a.e. in [t.,t2] (see (1.8)). For x,p,v admissible we define the cost functional J(x,p,v) = g(tl,x(tl),t2, x(t2)). We have here a Mayer-type problem. (2.7.i) (a closure theorem). Any AC limit x(t), tl K t K t2, in the p-metric of generalized trajectories is a generalized trajectory. In particular, any AC limit of (usual) trajectories is certainly a generalized trajectory. In other words, if xk(t), Pk(t), vk(t), tlk < t - t2k, k = 1,2,.., is any sequence of admissible generalized systems, if x(t), tl < t < t2, is any AC function, and p(xk,x) - 0 as k o, then x is an admissible generalized system, that is, there are measurable functions p(t), v(t), t <K t < t2, such that x(t), p(t(t), t), tl < t < t2, is an admissible generalized system, and J[xk,pk,vk] ] J[x,p,v] as k - o. Note that, whenever we wish to disregard boundary conditions, we have only to take B = E2n+2' and g and J need not be defined. t2 For Lagrange-type problems with usual functional I = Jft f (t,x(t),u(t))dt replaced by the generalized functional J = t2 ho(t,x(t), p(t).v(t))dt (see (1.9)), we consider as in (2.6) the new state variable x = (x,x) I0 n t m (x,x,...,x ), the same usual control variable u = (u,...,u ), and generalized control p = (pl,...,p ), v = (U,.. Here we assume f (t,x,u) continuous on M, we restrict the class of admissible generalized systems to only those for which h (t,x(t),p(t),v(t)) is L-integrable in [tl,t2], we replace in the hypotheses above xkx for xkX, we assume now that the sets 16

R(t,x) = co Q(t,x) = h(t,x, r U(t,x) ) are closed, and satisfy property (Q) with h = (h,h) = (h,h1,...,hn), and we assume that p(xk,x) + 0 as k +X. Statement (2.7.1) holds now with no further changes for Lagrange-type problems. (b) Instead of the hypotheses stated at the beginning of (a), we shall now assume only that A is any compact set of the tx-space E1+, that the set M is also compact, and that B is closed and g is continuous on B as before. We consider again the Mayer-type problem J[x,p,v] = g(tl,x(tl),t2,x(t2)). (2.7.ii) (a compactness theorem). The family of all generalized trajectories is sequentially compact in the p-metric. Indeed, here A is compact, M compact, hence N is compact, and the proof proceeds now as for (2.6.ii) with R replaced by Q. For Lagrange-type problems, with usual functional I = ft2 f (t,x(t),u(t))dt t, 0 replaced by J = ft2 h (t,x(t),p(t),v(t))dt, f continuous on M, no other hypothesis is needed. Thus, for A compact, M compact, B closed, statement (2.7.ii) holds for the family of generalized trajectories x(t) = (x,x) (xO,x,...,x ), tl < t < t2. Note that here the sets R(t,x) are all compact, metrically upper semicontinuous, and since they are convex by definition, they certainly have property (Q), as for the sets Q in (2.6.b). Part (A) above alone will be used in Chapter 4 (problems with bounded controls). Parts (2.B), (2.C), (2.D) below will be used only in Chapter 5 (problems with unbounded controls). 17

2.B. CLOSURE THEOREMS FOR FUNCTIONS WITH SINGULAR COMPONENTS 2.8. A CLOSURE THEOREM FOR FUNCTIONS WITH SINGULAR COMPONENTS We shall need a variant of Closure Theorem 1. We shall assume here that the space E is actually a product space E x E, hence x = (y,z) with y c E, z E. Analogously, we shall assume that A is a subset of the ys n-s o space E, and we shall take A of the form A = A x E, so that we have as s o n-rs usual A C E. We shall finally assume that the orientor field in A has the form n dx/dt E Q(t,y), (2.8.1) in other words, the set Q depends on t and y only, and not on z. A solution of this orientor field is then an AC n-vector function x(t) = (y(t),z(t)), tl < t < t2, with (t,x(t)) E A, that is, (t,y(t)) E A for every t e [tl,t2], and dx/dt e Q(t,y(t)), a.e. in [tl,t2] that is (y'(t),z'(t)) E Q(t,y(t)) (2.8.2) (2.8.i) Closure Theorem 2. Let A be a closed subset of the ty-space E1 x Es, and A = A x En so for every (t,y) E A let Q(t,y) denote a closed s o n-s0 0 subset of E, and assume that the sets Q(t,y) are convex, closed, and have property (Q) at every point (t,y) e A with exception perhaps of a set of points whose t coordinate lies on a set H of measure zero on the t-axis. Let xk(t), tlk < t < t2k, k = 1,2,..., be a sequence of solutions of the orientor field 18

(2.8.1), 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-s)-vector zk(t) converges pointwise for all t, 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 [tl,t2], and S!(t) = O a.e. in [tl,t2], that is, S(t) is a singular function. Then, the AC n-vector X(t) = [y(t),Z(t)], tl < t < t2, is a solution of the orientor field (2.8.1). In other words, we know that each xk(t) = (k(t),zk(t)), tlk < t < t2k, k = 1,2,..., is AC, that (t,yk(t)) E A for every t E [t lk,tk ] and that (y'(t),z'(t)) E Q(t,yk(t)) a.e. in [tlkt2k], we know that p(yk,y) + 0 hence tlk + t1, t2k - t2 as k + o, that zk(t) + z(t) = Z(t) + S(t) for every t c (tl,t2), that S(t) is singular, and (y(t),Z(t)) is AC, and we want to prove that (t,y(t)) E A for every t E [tl,t2] and that (y'(t),Z'(t)) c Q(t,y(t)) a.e. in [tl,t2]. For s = n this Closure Theorem 2 reduces to Theorem 1. Proof of (2.8.i). The proof that (t,y(t)) E A for every t E [tl t2] is the same as for Closure Theorem 1. Let us prove the remaining part of (2.8.i) where we shall need to know only that zk(t) - z(t) for almost all t e (tl,t2). For almost all t E [tl,t2] - H 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 t be such a point with tl < t < t2. Then, there is a a > O with tl < to - a < to + a < t2, and, for some k and all k > k, also tlk < t t + a < t Let x = X(t ) = (yZ), or yo = y(t ), Z = Z(t ). Let z = z(t ), S = S(t ). We have S'(t ) = O, hence z'(t ) exists and z'(t ) = Z'(t ). Also, we know that zk(t ) + z(to) 19

We have yk(t) + y(t) uniformly in [t - a,t + 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 > O, there is S > 0 such that t,t' E [t - a, t + a], It - t' I < 5, k > k implies 0 - 0 ly(t) - y(t')I < S/2, ly k(t) - Yk(t')J I< </2 We can assume 0 < 5 < a, 5 < ~. For any h, 0 < h < 5, let us consider the averages mh = h-l h X'(t + s)ds = h-l[X(t + h) - X(t )], (2.8.3) 0 0 0 mhk = h f xk(t + s)ds = h l[xk(t + h) - xk(t )], (2.8.4) where X = (y,Z), xk = (Yk,Zk) Given r > 0 arbitrary, we can fix h, 0 < h < 5 < a, so small that imh - X'(t)I < B, (2.8.5) IS(to + h) - S(to)) < 1 h/4. (2.8.6) This is possible since hl fh X'(t + s)ds + X'(t ) and [S(t + h) - S(t )]h-1 0 as h + 0+. Also, we can choose h in such a way that zk(t + h) z z(t ) as k - +o. This is possible since zk(t) + z(t) for almost all tl < t < t2. Having so fixed h, let us take kl > ko so large that jYk(t ) - Y(t ) I lYk(t + h) - y(to + h) I < min[t1 h/4, S/2], Izk(to) - z(t)J, Izk(t + h) - z(t + h)l < ~ h/8 20

This is possible since Yk(t) + Y(t), zk(t) > z(t) both at t = t and t = t + h. Then, we have Ih-l[yk(t+ + h) - Yk(t o) - h-[y(t + h) - y(t)] < Ih-[yk(t + h) -y(t + h)]I + Ih-[Yk(to) - y(to)]l < h-l[r h/4 + h-l(p h/4) = /2. Analogously, since z = Z + S, we have Ih-[zk (t + h) - zk(to) - h-'[Z(t + h) - Z(t )]l = Ih-l[zk(t + h) - zk(t )] - h- [z(to + h) - z(t )] + h-'[S(to + h) - S(t )]I < Ih-l[zk(t + h) - Z(to + h)]l + Ih-[zk(to) - z(to)]l + Jh-'[S(t + h) - S(t )]| < h'l( ( h/8) + h'l(T h/8) + h-l(r h/4) = q/2 Finally, we have lmhk - mhl = Ih- [xk(to + h) - xk(to)] - h-'[X(to + h) - X(to)]l Ih [yk(t +h) - yk(to)] - h-'[y(to + h) - y(to)]l + Jh-[zk(to + h) - zk(t0)] - h-l[Z(t + h) - Z(t )]J < q/2 + rI/2 = q. (2.8.7) 0 0 We conclude that for the chosen value of h, 0 < h < 5 < a, and every k > kl we have Ih - X'(to)J < r, hk- mhI < rl, IYk(t ) - y(t )I < E/2. (2.8.8) 21

For 0 < T < h we have now lYk(to + r) - y(to) < Yk(t + ) - Yk(to) + Yk(to) - y(t) I /2 + /2 = J(to + T) - tot < h < b < E, (2.8.9) x'(t + ) = (Yk(t + ), z'(t + L)) e Q(to + T, yk(t + T)) a.e. in [0,h]. ko ko ko o ko Hence, for almost all t, 0 < T < h, xk(t + ) = (Yk(to + o ),zk(t + T)) E Q(toYo,2), and consequently xk(t + T) = (Yi(t + T),zk(t + T)) E Cl co Q(to,y o,2) a.e. in [O,h]. ko 0 The average mhk as defined by (2.8.4) is then also a point of the same closed and convex set, or Mhk E cl co Q(to,yo,2~) for the chosen h and every k > kl. By relations (2.8.5) and (2.8.8) we deduce Jx'(to) - mhkl < jx'(to) - mhl + Imh - mhk and hence X'(t ) E [cl co Q(to,Yo,2~)]2 Here q > 0 is an arbitrary number, and the set in brackets is closed, hence X'(to) E cl co Q(toYo, 2~) 22

for every ~ > O. By property (Q) we have X'(t ) e nf cl co Q(to, Yo, 2~) = Q(to,y Y), where yo = y(to), and X'(to) = (y'(to),Z'(to)). We have proved that for almost all t E [tl,t2] we have dX/dt E Q(t,y(t)) Closure Theorem 2 is hereby proved. The following example illustrates Closure Theorem 2. Let n = 2, s = 1, n - s = 1, A = E3, Q = Q(t,y) = [z = (z1, z2)z2 > O _1 Z1 1]. If CP(t), 0 < t < 1, denotes a singular continuous monotone function with cp(O) = 0, cp(1) = 1, cp'(t) = 0 a.e. in [0,1], let us define p(t) in (-a,+oo) by taking t+k'l cp = 0 for t < 0 and cp = 1 for t > 1. Let zk(t) = k f CP(T)dT, 0 < t < 1, k = 1,2,.... Here the scalar functions zk(t) are absolutely continuous, monotone nondecreasing, with zk(t) > O and zk(t) +z(t)= ip(t) uniformly in [0,1] as k - a. Let us take Z(t) = O, y(t) = O, Yk(t) = O0 0 t < 1, k = 1,2,..., and then z(t) = Z(t) + cp(t), Z(t) absolutely continuous, cp(t) singular. Here (YkZk) converges uniformly toward (y,z) in [0,1]. All pairs (yk,Zk) are solutions of the orientor field (y',z') e Q, (y,Z) is a solution of the same orientor field, but (y,z) is not. Remark 1. We could now deduce from (2.8.i) corollaries similar to the closure statements (2.6.i) for usual solutions, and (2.7.i) for generalized solutions. This task can be well left to the reader as an exercise. The essential point is that while for usual solutions we need require explicitly that the sets Q, or Q, are convex, for generalized solutions the corresponding sets R are necessarily convex. 2

2. C. LOWER CLOSURE THEOREMS 2.9. LOWER CLOSURE OF FUNCTIONALS IN INTEGRAL FORM We shall now introduce the concept of lower closure. As usual let A be a closed subset of the tx-space E1 x En, for every (t,x) E A let U(t,x) be a given 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) = (fo,fl,...,fn) = (fo,f) be a given continuous vector function on M. Let B be a closed subset of the tlxlt2x2-space E2n+2 As in (1.2) and (2.6, (b)) we consider the func2n+2 tional I[x,u] = ft2 fo(t,x(t),u(t))dt. (2.9.1) As in (1.1), (1.2) we shall say that a pair x(t), u(t), tl < t < t2, is admissible provided x(t) is AC in [tl,t2], u(t) is measurable in [tl,t2], (t,x(t)) c A for t E [tl,t2], u(t) c U(t,x(t)) a.e. in [tl,t21, dx/dt = f(t,x(t),u(t)) a.e. in [tl,t2], f (t,x(t),u(t)) is L-integrable in [tl,t2], and (tl,x(tl),t2,x(t2)) E B. Whenever we wish to disregard boundary conditions, we have only to take 2n+2E Let x(t), tl < t < t2, be any AC vector function (which is the limit in the metric p of admissible trajectories). If, for any sequence xk(t), uk(t), tl t < t2k, k = 1,2,..., of admissible pairs with p(xk,x) + O, lim I[xk,uk] < + oo as k -+ oo, there is some measurable function u(t), t6 < t < t2, such that x(t), u(t), tl < t < t2, is admissible, and I[x,u] < lim I[Xk,Uk], (2.9.2) k-o 24

then we say that I[x,u] has the property of lower closure at the trajectory x(t), tl < t < t2. Before we prove a sufficient condition for lower closure, the following remarks are needed. First, if x is the limit in the p-metric of admissible trajectories as assumed, then by (2.1.i), we know that (t,x(t)) E A for all t E [tl,t2], and (tl,x(tl), t2,x(t2)) E B. Furthermore, if we know that the set M is closed, and that for every (t,x) E A the sets Q(t,x) = f(t,x,U(t,x)) are closed convex subsets of E satisfying property (Q) in A, then certainly x'(t) e Q(t,x(t)) a.e. in [tl,t2] by force of Closure Theorem 1 (2.5), and then there is some measurable u(t), t_ < t < t2, such that u(t) E U(t,x(t)), x'(t) = f(t,x(t),u(t)) a.e. in [tl,t2], (2.9.3) by force of Implicit Function Theorem (1.6.i) as we have seen in (2.6.i). As usual, we say that any such strategy u(t) generates x(t), tl < t < t2. Obbiously, in the comcept of lower semicontinuity we require more, namely we need a strategy u generating x for which (2.9.2) holds. It may well occur that x is generated by some strategy u for which (2.9.2) does not hold. The following example displays two strategies u and u, both generating the same trajectory x, such that (2.9.2) holds for u but not for u. Indeed, take m = n = 1, tl = 0, t2 = 1, f = 1 + cos Itu, f = fl = sin iu, u c U = [-1 < u < 1], x(t) = 0, 0 < t < 1, A = E2. Now take uk(t) = +2-1 according as k-i t < k-li + (2k), or k-li + (2k)- t (i + )ki = O,l,...,k - 1, k = 1,2,..., and take xk(t) = t - kI-li, or xk(t) = k-l(i+l) 25

according as t is in one or the other set of intervals above. Then Xk, uk, k = 1,2,..., is a sequence of admissible pairs, 0 < xk(t) < (2k)-1, and xk * x as k * o uniformly in [0,1]. The trajectory x(t) = 0, 0 < t < 1, is now generated by both u(t) = 1, 0 < t < 1, and by u(t) = 0, O < t < 1. On the other hand, I[x,u] = O, I[x,u] = 2, I[xk,uk] =, k 1,2,..., and thus relation (2.9.2) holds for u but not for u. As we shall see in (2.13), the concept of lower closure introduced above contains as a particular case the usual concept of lower semicontinuity, in particular the concept of lower semicontinuity for free problems. 2.10. A SUFFICIENT CONDITION FOR LOWER CLOSURE Let A, U(t,x), M, B, f (t,x,u), f(t,x,u) = (fl,...,fn) be defined as in (2.9). For any (t,x) E A let Q(t,x) be the set of all z = (z~,z,...,zn) =(z,z) with z0 > f (t,x,u), z = f(t,x,u) for some u E U(t,x). (2.10.i). If the sets A, M, B are closed, and f (t,x,u), f(t,x,u) = (fl,.,fn) are continuous on M, let us assume that the sets Q(t,x) are closed, convex, and satisfy property (Q) at every point (t,x) E A with exception perhaps of a set of points whose t coordinate lies on a set of measure zero on the t-axis. Let us assume that, for some locally L-integrable scalar function t(t) we have (4r)f (t,x,u) > V(t) for all (t,x,u) E M, with exception perhaps of another set of points whose t coordinate lies on a set of measure zero on the t-axis. Then the integral (2.9.1) has the property of lower closure at every AC vector function x(t) = (xl,...,xn), t1 < t < t2, which is the 26

limit in the p-metric of admissible trajectories. In other words, for every AC vector x(t) = (xl,...,xn), tl < t < t2, and sequence xk(t), uk(t), tlk < t < t2k, k = 1,2,..., of admissible pairs with p(xk, ) + O, lim I[xk,uk] < + oo as k -* o, there is a measurable function u(t), tl < t < t2, such that x(t), u(t), tl < t < t2, is admissible and I[x,u] < lim I[xk,uk]. Remark 1. Condition (r) in statement (2.10.i) will be drastically reduced in statement (2.10.ii) below. A simple condition under which the sets Q above, if convex, are closed and satisfy property (Q) as requested, and under which also condition (4r) above is satisfied, will be given in (2.12.i). Proof of (2.10.i). As usual we introduce auxiliary variables x~ and u~, vectors x = (x0 xl..1 xn), u = (uo,ul,...,um), and vector function f(t,x,u) = (fofl1...' fn) = (f0,f) with fo = uO. Let U(t,x) be the control space [u = (uO,u)lu~ > fo(tx,u), u E U(t,x)] c E+1. Then Q(t,x) = [z = (z~0,Z)1z > fo(t,x,u), z = f(t,x,u), u E U(t,x)] (2.10.2) f(t,x,U(t,x)) while Q(t,x) = [zlz = f(t,x,u), u E U(t,x)] = f(t,x,U(t,x)). We have now an auxiliary canonic problem with dx/dt = f(t,x,u), u e U(t,x), (t,x) E A, x~(tl) = O, (tl,x(tl),t2, x(t2)) E B, 27

and functional J t2 uO(t)dt = x (t2) tl Here p(xk, x) - 0 as k - om, hence tlk - tl t2k - t2, and thus tl - I < tlk < t2k < t2 + I for all k = 1,2,..., and some constant I > 0. Finally -t2k I[k k = t2k f(txk(t),uk(t))dt >- ft2+Q t(t)dt = L lk where L > 0 is now a fixed number. Let i = lim I[xk,uk], so that -L < i < + o. Let us assume i < + o, and k-oo let us consider a subsequence [k ] such that I[xk,uk ] i as s -> o. Let L1 s s be a constant such that I[xk,us] < L1 for all s. s Now we have o. t xk(t) = t fO(T,xk(T),uk() )d (2. 10.3) t t t I [f (TXk(T),Uk(T)) + ] - r(T)dT 1k tlk where f + >r 0, and thus if Fk (t) =j [f (T, Xk(T),Uk(T) + jr(T)]dT, t < t<t (2.10.4) we have 0 < Fk (t) < L1 + L 28

for all s, and Fk (t) is a nondecreasing continuous function in [tlk,t2k]. s We shall actually extend these functions in the fixed interval [tl - L, t2 + I] by continuity and constancy outside [tlk,t2k]. By Helly's theorem there exists a subsequence, say still [k ] for the sake of simplicity, such that Fk (t) * F(t) as s -* pointwise in [tl - I, s t2 + ]. Then F(t) is a monotone nondecreasing function in [tl - 2, t2 + ], with 0 < F(t) < L1 + L. Since x (t) = Fk (t) - ft r(T)dT, (2.10.5) ks s lk we conclude that xk (t) * x0(t) pointwise in (t1,t2), and that k x (t) = F(t) - Jfi t(T)dT, tl < t < t2 (2.10.6) tl Note that the sequence x~(t2) is bounded, hence we can extract the subsequences 2k (k ) above, so that the limit x2 = lim x (t2k ) exists. Let us define x~(t) Soo S at tl and at t2 by taking x (tl) = 0 and x (t2) = Let us prove that 0 0 0 0given E >, we x (t) = 0 < x (tl + 0) and x~(t2 - 0) < x2 x (t2). Indeed, given ~ >, we have Fk (tlk ) = 0 < Fk (tl + E), and, as s + +, also 0 < F(tl + ~). Finally, S s S as E - O, we have 0 < F(tl + 0), and by (2.10.6) also x (tl) = 0 < x0(tl + 0). Analogously, we have t2 - E < t2k for k sufficiently large, hence Fk (t2 - ~) < K Fk (t2k ), hence, by (2.10.3), and (2.10.4), also s S 29

xk (t2f ~) + t A(T)dT < Xk (t2k + t s t(T)dT s lk s s lk S S As s - + oo we have x0(t2 - E) + ft2- r(T)dT < x + ft2 (T)dT and, as E + O +, also x (t2 -) _<x = x (t2) Now F(t) = xo(t) + f 4 (T)dT is a monotone, nondecreasing, nonnegative function in [tl,t2] with F(tl) = 0, and hence F possesses a unique decomposition F = F* + S into an AC function F* and a singular function S, both F*, S monotone, nondecreasing, nonnegative functions in [tl,t2] with F*(tl) = S(tl) = O. If now we set X(t) = F*(t) - f t (T)dT, tz < t < t2, t1 we have F(t) = x (t) + fI (T)dT F*(t) + S(t) x (t) = (F*(t) - f r(T)dT) + S(t), or 3o

x (t) = X(t) + S(t), where X is AC and S(t) > 0 is monotone, nondecreasing, and singular. Let x(t) = (X,x), tl < t < t2. Then dX/dt = dx /dt, X(tl) = O. By Closure Theorem 2 we conclude that x'(t) = (X'(t),x'(t)) E Q(t,x(t)) a.e. in [tl,t2], and by Implicit Function Theorem (l.6.i) there is a measurable function u(t) = (u,u), tl < t < t2, such that u(t) EU,xX()), x'(t) = f(t,x(t),u(t)), or u (t) > f (t,x(t),u(t)), u(t) E U(t,x(t)), dX/dt = u (t), dx/dt = f(t,x(t),u(t)) a.e. in [tl,t2]. On the other hand I[xu] Jft2 f (t,x(t),u(t))dt < f u~(t)dt tJ 0 t0 X(t2) - X(tl) = X(t2) = X (t2) - S(t2) 0 0 0 0 < x(t2) = = lim xk (t2k S-oo S S lim I[xk,uk ] = i = lim I[xk,uk] S-+oo s s k+oo Remark 2. In the sufficient condition for lower closure (2.10.i) it is enough 31

to request that the sets Q(t,x) are closed, convex, and have property (Q) at the points (t,x(t)) E A for almost all t E [tl,t2]. In this form, and under suitable regularity hypotheses, the convexity assumption of this sufficient condition for lower closure will be shown to be also necessary [App. A.5]. Remark 3. Condition (sr) in statement (2.10.i) is certainly satisfied if, for instance, f (t,x,u) > 0 for all (t,x,u) E M, or f (t,x,u) > v for all (t,x,u) E M where v is some real constant. Nevertheless, condition(,f) in (2.10.i) can be reduced. For instance, we may replace it by the following weaker assumption (&'): for every compact subset A of A there is a locally integrable function Iro(t) (which may depend on A ) such that f (t,x,u) > If(t) for all (t,x) C A, u E U(t,x). The proof is the same since we can include all trajectories x and Xk in a unique compact subset A of A. k 0 A more drastic generalization of (2.10.i) will be given below (2.10.ii) where we shall use the following much weaker form of condition. Condition (g*). For every (t,x) e A there are a neighborhood N(t,x) of (t,x) in A, a locally integrable function jr(t), and real numbers bl,...,bn (all bl,...,bn and Ar may depend on t,x,N) such that f(t,x,u) = fo(t,x,u) - Zn b f (t,x,u) > 4(t) f~tJX'9u) o j j fj~t'xp - u)-= for all (t,x) e N(t,x), u E U(t,x), with exception perhaps of a set of points (t,x) whose t coordinate lies on a set of measure zero on the t-axis. Remark 4. We shall note here that, under condition (t*), it is natural to 32

consider the sets Q(t,x) = [(z,z)lz > f (t,x,u), z = f(t,x,u), u E U(t,x)], or the analogous sets Q*(t,x) = [(Z,Z)IZ > f (t,x,u), Z = f(t,x,u), u e U(t,x)] It is easy to see that the sets Q are closed, or convex, or satisfy property (Q) if and only if the same occurs for the sets Q*. Indeed, the sets above are transformed into one another by the fixed affine transformation O O Z = z - bbz, Z = z (2.10.ii). Let A, B, U(t,x), M, f(t,x,u), f (t,x,u) as in (2.11.i), and let us assume that condition (4*) holds. With N(t,x) as in condition (t*), and for every (t,x) E N(t,x), let Q(t,x) denote the set of all z = (zz,... n) = (z,z) with z > f (t,x,u),z = f(t,x,u) for u E U(t,x), and assume that the sets Q(t,x) are closed, convex, and satisfy property (Q) at all points (t,x) E N(t,x), with exception perhaps of a set of points whose t-coordinate lies on a set of measure zero on the t-axis. Then the integral (2.9.1) has the property of lower closure at every AC vector function x(t), tl < t < t2, which is the limit in the p-metric of admissible trajectories. Proof. Let x(t), tl < t < t2, be any AC function as in text, and xk(t), uk(t), tlk < t < t2k, k = 1,2,..., be a sequence of admissible pairs with P(Xk,X) + 0, lim I[xk,uk] < + <. Let A be a compact neighborhood (containing

the graph of x and all Xk). By hypothesis, for every (t,x) E A there are numbers 5 > 0, bl,...,bn real, and a locally integrable function *(t), - co < t < + o, such that f (t,x,u) ~> (t) for all (t,x) E N 2(tx) and u c U(t,x). We consider the smaller neighborhoods N6(t,x) which we consider as open (in A). These too form a cover of the compact set A. Thus, finitely many of these N6 cover A, say N8 (t,x ), 7 l,...,s. Let 5 > 0, b 1 Y b, 4 be the corresponding elements, so that yn y f (txu) = f (t,x,u) -Z b fj(t,x,u) > (t) 05 0 j=l j, 7 for all (t,x) N25 (t x ), u E U(t,x), y = l...,s, and UsN (t7 A 7 7 Let b = max [lb I, j = l,...,n, y = l,...,s], = min [, = l...s]. Since (t,x(t)) E A for all tl < t < t2, we can divide the arc C: x = x(t), tl < t < t2, into finitely many subarcs, say C, a = 1,...,N, each C completely contained in some neighborhood N8 (ty,x7y) Thus, we have for the 7 arcs C the representations C: x = x(t), T < t < Tr with t1 = To < Tj <... < TN = t2, and each Ca lies in a certain N6 (t,x ) which now remains associated to C. Since p(xk,x) - 0 as k >+ o, hence tlk + t1, t2k - t2, we see that for all k sufficiently large we have tlk < T1 <... < TN_1 < t2k. Thus for all k sufficiently large, the arc Ck: x = xk(t), tlk < t < t2k, is divided into the same number N of subarcs, say Ck: x = xk(t), Ta1 < t < T, a = 1,...,N, where now T = t1 must be replaced by tlk and TN = t2 must be replaced by t2k. Also, for all k sufficiently large, say for k > ko, the arc Ck is completely contained in N26 (t,x ) for the same y we have already 34

associated to C. Thus, for k > k, C lies in some N5 (t,x ) and C in a- o 6 Y kac N26 (t,x ). Also, Ckc + C as k -+ 0 in the sense that p-distance approaches zero as k -+. We shall now consider for each a = 1,...,N, the auxiliary functional t - J = t f (t,x(t),u(t))dt t' 0 for all admissible pairs x,u with the graph of x lying in N25 (t,x ). Here Y by admissible we mean that the conditions a-d of (1.1) are satisfied with A replaced by N25 (t,x ), and of course f (t,x(t),u(t)) L-integrable as usual. For each a we may now apply (2.11.i) to arc C, the sequence Ckc, k = 1, 2,..., and functional J. We conclude that each C is admissible and that J[C ] < lim J[C k], a = 1,...,N. (2.10.7) k-e a More precisely, for each a, there is a measurable u(t), T < t < T, such a-l- - a that the pair x(t), u(t), T1 < t < T is admissible for the functional J, in particular u(t) e U(t,x(t)), dx/dt = f (t,x(t),u(t)), T1 < t < T (a.e.), a = 1,...,N, and the expression fo(t,x(t),u(t)) = f (tx(t),u(t)) - j=l b j fj(t,x(t),u(t)) is L-integrable in [T 1,T ]. Since the functions fj here are certainly Lintegrable in the same interval (as derivatives of the AC functions x (t) in [T,T ]), we conclude that f (tx(t),u(t)) itself is L-integrable in each [T 1,T ] and hence in the whole of [tl,t2]. We have proved that the pair x(t), 5

u(t), tl < t < t2, is admissible for the original integral I. Now, given ~ > 0, we deduce from (2.10.7) that there is some k > k such that, for k > k, we have J[Cka - J[C ] > - E/N, p(Cka,C ) < E/Nnb, o = 1,...N (2.10.8) Now we have I[x,u] = ft2 fo(tx(t),u(t))dt = N fTa f (t,x(t),u(t))dt tj 0 a=l T 0 zTTa f (t,xk(t)uk(t))dt + Zn b [xa(T k) alLJT k kj=1l j k -1 (2.10.8), where we have written bnow in abstead of value < (Nnby the index we have associated to z[Xk,%]k- T[x,u] >- N(s/N) + + En b [x (T) -Xj(T )+ [X T (T 56

all k > k we have I[xk,Uk] - I[x,u] > - - Nnb[2(Nnb)-] = - 3~ Because of the arbitrarity of e, we have proved the lower closure of I at x. Remark 5. Statements (2.10.i) and (2.10.ii) have corollaries for generalized solutions. Let A, U(t,x), M, B, fo(t,x,u), f(t,x,u) = (f,...,f) be defined as in (2.9). Thenx(t), p(t), v(t), tl < t < t2, is said to be an admissible generalized system (generalized solution) provided x(t) is AC and p(t), v(t) are measurable in [tl,t2]; (t,x(t)) E A for all t E [tl,t2]; p(t) = (Pl,-..,Py), pj(t) > 0, Ejpj(t) = 1 (that is, p(t) e r), v(t) = (u((t), j =,...,y), u(j)(t) C U(t,x(t)) a.e. in [tl,t2]; provided the differential equation dx/dt = 7_ p (t) f(t,x(t),u(j)(t)) j=1 j is satisfied a.e. in [tl,t2], the function Z. 1 p f (t,x(t),u(j)(t)) is Lintegrable in [tl,t2], and (tl,x(tl),t2,x(t2)) E B. Then the corresponding functional is I[x,p,v] = t =l pj(t) (t),x(t),u(j)(t))dt The sets R(t,x) of all points z = (z~,z) = (zO,z,..., n) with zO > Z p f (t,x,u(j)), z = Z pj f(t,x,u(j)) for some (p,v) e r x (U(t,x)7 are exactly the sets co Q(t,x) if 7 = n + 2. We take for 7 the smallest integer for which this occurs for all (t,x) E A, 1 < y < n + 2. (2.10.iii). (A lower closure theorem for Lagrange problem and generalized solutions). If the sets A, M, B are closed, and f, f are continuous on M, 37

let us assume that the convex sets R(t,x) are closed and satisfy property (Q) at every point (t,x) c A with exception perhaps of a set of points whose t coordinate lies on a set of measure zero on the t-axis. Let us assume that for some locally L-integrable scalar function *(t) we have (4r) f (t,x,u) >,r(t) for all (t,x,u) C M, with exception perhaps of another set of points whose t coordinate lies on a set of measure zero on the t-axis. Then the integral (2.10.7) has the property of lower closure at every AC vector function x(t) n = (xl,...,x ), tl < t < t2, which is the limit in the p-metric of generalized trajectories. In other words, for every AC function x(t), t1 < t < t2, and sequence, xk(t), Pk(t), vk(t), tk < t < t2k k = 1,2,..., of generalized systems with p(xkx) - 0 as k - o, lim I[x kpkvk] < as k +, there are measurable functions p(t), v(t), ti < t < t2, with p(t) = (P1,.,p ), p (t) > 0, p(t) = 1 v(t) = (u(j)(t),j = 1,...,), u(j)(t) c U(t,x(t)) a.e. in [tlt2], Cpj~t)=l, J such that I[x,p,v] < lim I[xk,pkvk] We leave to the reader to state the analogous corollary of (2.10.i) for generalized solutions. 2.110 A VARIANT OF THE LOWER CLOSURE PROPERTY Statement (2.10.i) holds in a slightly stronger form. To formulate it we need, besides the sets Q(t,x) c E of (2.10), also the sets Q(t,x) = f(t,x, n+l U(t,x)) c E. These sets Q(t,x) are the projections on the z-space E of the n n sets Q(t,x) of the z~z-space E +1. Thus, if the sets Q(t,x) are convex, so are the sets Q(t,x). On the other hand, the sets Q(t,x) may be closed, without the sets Q(t,x) being so. This is shown by the example n = 2, m = 1, U = =[- ~ < u < + a], f = (1 + u2)i/2, f = arctan u, - r/2 < f < i/2. Then, Q 38

and Q are the fixed sets Q = [zi - r/2 < z < it/2] c El, Q [(z,z)I z > sec z, - K/2 < z < K/2] c E2, and Q is closed, but Q is not. This example shows also that property (Q) for the sets Q does not imply the same property for the sets Q. In the statement below we shall require that both the sets Q(t,x) and the sets Q(t,x) have property (Q). (2.11. i). If we assume, in addition to the hypotheses of (2.10. i), that both the sets Q(t,x) c En+1 and the sets Q(t,x) c E are closed, convex, and satisfy property (Q) at all points of A with exception perhaps of a set of points whose t coordinate lies on a set of measure zero on the t-axis, then for every sequence xk(t), u(t), tlk < t t2k, k = 1,2,..., of admissible pairs, and any AC vector function x(t) = (xl,...,xn), tl < t < t2, with p(xkx) - O as k _ oo, there is a measurable function u(t), tl < t < t2, such that u(t) E U(t,x(t)), x'(t) = f(t,x(t),u(t)) a.e. in [tl,t2], and I[x,u] < lim I[xk,uk]. (2.11.1) k-oo If lim I[xk,uk] < + o, then certainly the pair x,u is admissible, and (2.11.1) holds. An analogous variant of theorem (2.11. i) also holds. Proof of (2.1l.i). First note that f (t,x(t),u(t)) is measurable in [tl,t2] and > r(t), hence, I[x,u] is finite, or + oo. If the second member of (2.11.1) is finite, then I[x,u] must be finite, hence f (t,x(t),u(t)) must be L-integrable, and the conclusion of (2.11.i) reduces to the conclusion of (2.10.i) in the case under consideration. If the second member of (2.11.1) is + 00, then (2.11.1) in itself is trivial, but we still have to prove that a measurable 39

u(t), ti < t < t2, exists with u(t) E U(t,x(t)), x'(t) = f(t,x(t),u(t)) a.e. in [tl,t2]. This, however, is a consequence of closure theorem 1 of (2.5) applied to the AC n-vector function x, the n-vector function f, and the sets Q(t,x) c E. Statement (2.11.2) is thereby proved. n Finally, let us show by an example that an integral I[x,u] may possess the properties of statement (2.10.i), and thus the property of lower closure as defined at the beginning of (2.9), and yet not possess the stronger property of the present section (2.11). Indeed, take m = n = 1, U = [ul - Co < u < + 00], f = exp(u), f = exp(u2), A = E2, and take x(t) =, x(t) = t < x(t) = k 0 < t < 1, k = 1,2,... Here xk + x uniformly in [0,1] as k - oo and I[xk,uk] = exp(log k)2 + + oo as k - + oo. Obviously, there is no measurable u(t), O < t < 1, with - co < u(t) < + oo, such that 0 = x'(t) = exp(u(t)) a.e. in [0,1]. The integral I has not the strong property represented by the conclusion of statement (2.11.i). Yet the integral I has the property of lower closure as defined in (2.9) as a consequence of theorem (3.6.i). Indeed, here Q = [z > exp(u2), z = exp(u), u E El], or Q = [z > exp(log z)2, 0 < z < + oo], is a fixed closed convex subset of E2, and all conditions of (2.10.i) are satisfied. Instead, Q = [z = exp(u), u e El] is the set Q = [O < z < + oo], a fixed convex set, and Q is not closed. 2.12. CRITERIA FOR PROPERTY (Q) OF THE SETS Q(t,x) We assume here that the sets A, U(t,x), M, Q(t,x), Q(t,x) are defined as usual, that the sets A and M are closed, and that the functions f (t,x,u), f(t,x,u) = (fl,...,fn) are continuous on M. 40

(a) We say that a function g(t,x,u) on M is of slower growth than f (t,x,u) as lul + o in a subset A of A if, for every 6 > 0 there is some number H, which may depend on E, f and A0, such that (t,x) E A, Jul > H, u E U(t,x) implies Igl < f. (2.12.i). If 1 and f are of slower growth than f as tul + m in a neighborhood N(t,x) of (t,x) in A, and Q(t,x) is convex, then the sets Q(t,x) satisfy property (Q) at (t,x) (in particular, Q(t,x) is closed). This statement is proved in (App. A.4.i). Note that, if 1 and f are of slower growth than f as Jul + co in A, then not only the sets Q(t,x) of (2.10.i) satisfy property (Q) in A, but also condition (4) of (2.10.i) is trivially satisfied with 1f = constant. Remark 1. As mentioned in (2.10), for generalized solutions of Lagrange problems we consider the convex sets R(t,x) = co Q(t,x), and we have to verify that these sets R(t,x) are closed, and satisfy property (Q). Statement (2.12.i) can be completed as follows: (2.12.ii). If 1 and f are of slower growth than f as Jul - oo in a neighborhood N (t,x) of (t,x) in A, then the convex sets R(t,x) = co Q(t,x) satisfy property (Q) at (t,x) (in particular, R(t,x) is closed). The statement is proved in (App. A.4. ). Thus, if 1 and f are of slower growth than f in A, then the convex sets R(t,x) of (2.10.iii) certainly are closed and satisfy property (Q) in A, and even condition (*) of (2.10.iii) is trivially satisfied (with f = constant). (b) In the criterion (2.12.iii) below we shall use a different set of hypotheses. At the beginning of (2.11) we noticed that the sets Q(t,x) c E n

are the projections of the sets Q(t,x) c En+1 on the z-space En; hence, the convexity of any set Q(t,x) in E+1 implies the convexity of the corresponding set Q(t,x) in E. Nevertheless, as we proved by an example (at the beginning of (2.11)) the sets Q(t,x) may be closed and even satisfy property (Q) at any given point (t,x) without this being the case for the sets Q(t,x). However, the following holds: if the sets Q(t,x) satisfy property (Q) at (tx), then (a) (z~,z) E n5 cl co Q(t,x) implies z E Q(t,x). Indeed (z,z) c nf cl co Q(t,x) yields (z,z) E Q(t,x) by property (Q) at (t,x), and then z e Q(t,x). We shall say that condition (a) holds at the point (t,x) e A provided: (a) (zO,z) E n cl co Q(t,x) implies z Q(t,x). As mentioned, this condition is necessary for property (Q) of the sets Q(t,x) at (t,x). This same condition (a) alone is not sufficient for property (Q) as the following example shows: Take m = n = U = E, f = t3U2, f = t f = tu, 0 < t < 1. Then Q(O) = [(z0,z) z0 >, z = 0], Q(t) = [(z,z) z0 > t- z2 z e E1] if t > 0, the sets Q do not satisfy condition (Q) at t = 0, but condition (a) certainly holds at the same point. Note that condition (a) is trivially satisfied for free problems (m = n, f = u, U = E ), since Q = U = E, and all points z E E are in Q. n Now we shall say that condition (X) holds at the point (t,x) E A provided: (X) For every z e Q(t,x) there is at least one point u E U(t,x) with z = f(t,x,u) and the following property: given E > O there are numbers b > 0, and r, b = (bi,...,bn) real, such that 42

(X') f (t,x,u) > r + Zj bj fj(t,x,u) for all (t,x) e Ns(t,x) and u c U(t,x); (X") f (txu) < r + j bj(txu) + F. As we shall see in (App. A. 4), this is a very weak requirement. For free problems, for instance, this condition reduces to a weak form of the well known "seminormal convexity condition" (App. A.6). (2.12.iii). If conditions (a) and (X) hold at a point (t,x) E A, then the sets Q(t,x) are closed, convex, and satisfy property (Q) at the point (t,x). This statement will be proved in (App. A.4. iv). We shall see there also, that under a slight requirement, the union of (a) and (X) is necessary as well as a sufficient condition for property (Q) of the sets Q(t,x). (c) The case of f linear in u. We shall assume here that A is a given closed subset of the tx-space E1+, that U =E, that f (t,x,u) and f(t,x,u) = +n m o (fl,.,f f n) are continuous on M = A x Em, and that f is linear in u, that is, n i fi(t,x,u) = J=1 bij(t,x)u + ci(t,x), i = or f(t,x,u) = B(t,x)u + C(t,x), where B, C are n x mand n x lmatrices with entries continuous in A. For every compact subset A of A, the functions bij, ci are continuous and bounded on A; hence, there are constants G, F such that If(t,x,u)l < Golul + F for all (t,x) E A and u e E. o n (2.12.iv). If f (t,x,u) is convex in u, and f linear in u with U = E, 43

then the sets Q(t,x) are convex. Proof. If ~ = (, = (,) are any two points of Q(t,x), and 0 < a < 1, let z = (z,z) = a + (1 - a)B. Then there are vectors u, v c E such that > f (txu), = Bu + C, T > f (t,x,v), r = Bv + C - 0 0 0 0 z = a + (1 - ), z = + (1 - a), z = a, + (1 - a) If w e E denotes the vector w = au + (1 - a)v, we have m z = 5 + (1 - ) = - (Bu + C) + (1 - a)(Bv + C) = B(au + (1 - a)v) + C = Bw + C z = at0o+ (a- )O a (txu)+ ( - a)~ > f (txu) + (1- )f(t,x,v) > fo(txCu + (- O)v) = fo(t,x,w) Thus z = (z,z) E Q(t,x) and Q(t,x) is convex. (2.12.v). If A is closed, U = E m M = A x E, if f (t,x,u) is continuous on M, convex in u, and "seminormal in u at a point x E A (see definition (SN) in (d) below), if f(t,x,u) = B(t,x)u + C(t,x), where the matrices B, C have entries continuous in A, then the sets Q(t,x) = [(z,z)lz > f (t,x,u), z = f(t,x,u),u ~ E ] satisfy property (Q) at (t,x). A proof is given in (App. A.4.(v)). This statement for f linear in u, or f = B(t,x)u + C(t,x), is much stronger than the analogous statement (2.12.i). Indeed, we would deduce from (2.12.i) an anologous statement as (2.12.iv) under 44

a growth condition f (t,x,u) > (|lI ul) with $(D)/ + + as 5 + + 00. 0 Example 1. Take m = 1, n = 2, U = El, f 1, fl = u, f2 tu, -1 < t < 1. 0 Then the sets Q and Q depend on t, and Q(t) = [z = (z'z2)Iz' = u, z2 = tu, u E El] [z = (z1 z2)1z2 = tz', Z 1 El] c E2, Q(t) = [z = (z 0 zz 2)z0 > 1, z2 = t1 z1 z E1] c E3 Each set Q(t) is a straight line in E2 of slope t, and for each 5 > 0 the set Q(O,5) contains both lines z2 = +~zl, and the convex hull of Q(0,5) coincide with the whole plane E2. Thus Q(O) is the z1-axis and nf cl co Q(0,5) is the whole zIz2-plane. The set Q(t) does not satisfy property (Q) at t = 0, and the same holds for Q(t). Here f = 1 does not satisfy the weak growth condition requested in (2.12.iv). Example 2. Take m = 1, n = 2, U = El, f = Itul, fl = u, f2 = tu, -1 < t < 1. Then Q(t) = [z = (z, z2)lz2 = t zl, zI E E1 c E2 Q(t) = [z = (zo,z,z2)zo > z2, z2 = tz, z C E ] E3 As before the set Q(t) does not satisfy property (Q) at t = O. Analogously, for any 5 > 0 and - 5 < t < 5, we see that z~' =(z0,1,z2 ) = (1,8 1,) E (8) z = (z0,z,z ) = (1, —,1) E Q(-8), 45

and for c = 1/2, also z = cz' + (1 c )z" = (zo,z~,z2) = (1,0,1) C coQ(O;8) Hence z = (1,o,1) = n cl co Q(O;5) z = (1,0,0) $ Q(O), and Q(t) does not satisfy property (Q) at t = O. Here f does not grow at t = 0. Example 3. Take m = 1, n = 2, U = E1, f = lul, f2 = tu, - 1 < t < 1. Then Q(t) = [z = (z', 2)1z2 = tz1, z1 e E1] c E2 Q(t) [z =(zo1,2)Izo > z, z2 = tZl, z) E E1] c E3 z As before Q(t) does not satisfy property (Q), while Q(t) does satisfy property (Q) at every t because of statement (2.12.iv). Example 4. Take m = n = 1, U = El, f = tu2, fl = u, O < t < 1. Then 0 Q(t) = [zlz = u, u E E1] c E1, Q(t) = [z = (z,z) z0 > tu2, z = = u, u E E1] c E2 Here Q(t) = U = E1 for every t, 0 < t < 1, and obviously Q(t) satisfies prop0 erty (Q). On the other hand, Q(O) is the half plane [z > O, z E EJ while Q(t) for t > 0 is the set Q(t) = [z0 > tz2, z e E1]. Obviously, Q satisfies property (Q) at t = 0 (and at every t as well). Example 5. Take m = n = l, U = El, f = t3u2, fl = tu, O < t < 1. Then 46

Q(t) = [zlz = tu, u E El] c E1, Q(t) = [z = (z,z)lz > t3u2, t = tu, u E El] c E2 Here Q(O) is reduced to the single point z = O, while Q(t) for every t > 0 coincides with El. Thus Q(t) does not satisfy property (Q) at t = O. Also, Q(O) = [z > 0, z = 0], while Q(t) for t > 0 is the set Q(t) = [z > t z2, z c E1] and cl co Q(0;5) is the entire half plane [z > 0, z E El]. Thus, neither Q nor Q satisfy property (Q) at t = 0. (d) The free problem case: m = n, f = u, U = E. Here the sets Q ren duce to the fixed, closed, and convex set Q = U = E. The sets Q(t,x) reduce here to Q(t,x) = [(z,u)|z > f (t,x,u), u E ]. These sets are closed whenever f is continuous, and convex whenever f (t,x,u) 0 0 is convex in u. As mentioned, condition (a) is trivially satisfied. Condition (X) at a point (t,x) E A reduces to the following simple (and well known) requirement: (Xf) ( = weak seminormality condition) For every u E E and E > 0 there are numbers 5 > 0, and r, b = (bl,...,bn) real such that (Xf) f (t,x,u) > r + b u for all (t,x) E Ns(t,x) and u E En; (Xf ) fo(t,x,u) < r + b.u + s Then statement (2.12.ii) yields: 47

(2.12.vi). For free problems (m = n, f = u, U = E ), if A is closed, if f (t,x,u) is continuous on M = A x E and convex in u, and if f is weakly seminormal at a point (t,x) E A, then the sets Q(t,x) satisfy property (Q) at (t,x). Convexity of f alone does not imply that the sets Q(t,x) have property (Q) in A. This is shown by the following simple example. Take n = 1, f (t,u) = tu, 0 < t < 1, u E U = E1. Then f is continuous and convex in u for every t, but at every t, 0 < t < 1, we have Q(t) = [(z,u)z0 > tu, u E El], a half plane in E2, while nA cl co Q(t;b) is the entire plane E2. Thus the sets Q do not satisfy property (Q) at anyt, 0_ <t < 1. The usual seminormality condition is a somewhat stronger requirement. By a nonessential modification of the condition used by L. Tonelli and E. J. Mc Shane we shall say that the seminormality condition at a point (t,x) E A is satisfied provided: (SN) (seminormality condition) For every u E E and ~ > 0 there are numbers 8 > O, v > 0, and r, b = (bl,...,b ) real such that (SN') f (t,x,u) > r + b u + vlu - uI for all (t,x) E N5(t,x) and u E En; n (SN") fo(t,x,u) < r + b u + ~. Seminormality condition has a very simple and elegant characterization: (2.12.vii). For free problems (m = n, f = u, U = E ), if f (t,x,u) is convex in u at some (t,x) E A, then f is seminormal at (t,x) if and only if for no u, ul E E, ul 0, it occurs that f (t,x,u) = 2-1 [f (t,x,u + Xul) + f0(t,x,u - \ul)] for all x > 0. 148

A proof of this statement is given in (App. A.6.i). Note that, if we denote by Q(t,x) the set [(z,u)1z0 = f (t,x,u),u e E ], then Q(t,x) is often denoted as the "figurative" of f (at the point (t,x) E A). Statement (2.12.vi) then states that f is seminormal at (t,x) if and only if the figurative con0 tains no straight line. In particular, if say f (txu) >+ oo as l ul + +o, 0 and f (t,x,u) is convex in u, then the figurative Q(t,x) cannot contain any straight line, f is seminormal at (t,x), f is weakly seminormal, and certainly the sets Q(t,x) satisfy property (Q) at (tZ,x). 49

2.D. LOWER SEMICONTINUITY THEOREMS 2.13. LOWER SEMICONTINUITY OF FUNCTIONALS IN INTEGRAL FORM The rather general concept of lower closure defined in (2.9) is a natural extension of the usual concept of lower semicontinuity. Indeed, the definition of lower closure in (2.9) reduces to the usual concept of lower semicontinuity whenever the strategy u is "determined" by the (admissible) trajectory x, and then the functional (2.9.1) can be thought of as depending on the (admissible) trajectory x only: t2 I[x] = f fo(t,x(t),u(t))d.t (2.15.1) This occurs, for instance, for free problems where u(t) = x'(t) (a.e.). Purpose of the present section (2.13) and next one (2.14) is to clarify the concepts, to deduce theorems of lower semicontinuity from our previous theorems of lower closure in (2.10) and (2.11)., and to show that the usual theorems of lower semicontinuity for free problems are corollaries of our theorems of lower closure. (a) As mentioned in (1.5) it may happen that the data A, U(t,x), B, f (t,x,u), f(t,x,u) = (fl-..,fn) are so arranged that, for any admissible pair x(t), u(t), ti < t < t2, the trajectory x determines uniquely the strategy u (a.e. in [tl,t2]). Then the functional (2.13.1) can be thought of as being defined for every admissible trajectory x, and we may denote it as I[x]. In (1.5) we referred to these systems as TDS systems. The free problems are in 5o

this class. For all these systems the concept of lower closure (2.9) reduces to the one of lower semicontinuity. Let x(t), tl < t < t2, be any AC vector function which is the limit in the p-metric of a sequence of admissible trajectories xk(t), tlk < t < t2k, k = 1,2,..., with p(xk,x) -o 0 and lim I[x k] < + 0 as k + m (thus, of course tlk t1 t2 t2). The functional (2.13.1) is said to be lower semicontinuous at x provided, from any such sequence we can conclude that x is admissible, and that I[x] < lim I[xk] as k + oo. (b) For general TDS systems statements (2.10.i) and (2.11.i) reduce to the following ones. (2.13.i). For systems TDS, and under the same conditions of (2.10.i) any AC function x(t), tl < t < t2, with (t,x(t)) E A for all t E [tl,t2] and which is the uniform limit of admissible trajectories xk(t), tlk < t t2k, k = 1, 2,..., with p(xk,x) + 0,lim I[xk] < + o as k oo, then x is admissible and I[x] < lim I[x k]. (2.13.ii). For systems TDS, and under the conditions of (2.10. i) and additional hypotheses in (2.11.i), any AC function x(t), tl < t < t2, with (t,x(t)) E A for all t E [tl,t2] and which is the uniform limit of admissible trajectories xk(t), tlk t t2k, k = 1,2,..., that is p(xk,x) + 0 as k - o, then x is admissible and I[x] < lim I[x k]. In the last statement we understand that there is a measurable function u(t), tl < t < t2, such that x(t), u(t), tl < t < t2, satisfyingall conditions for admissibility but perhaps the L-integrability of f (t,x(t),u(t)) in [tl,t2], and that this condition too is satisfied whenever lim I[xk] < + -o. Statement (2.10.ii) also has its counterpart here, but we leave its formulation to the reader. 51

2.14. THEOREMS OF LOWER SEMICONTINUITY FOR FREE PROBLEMS Let us consider here free problems, that is, systems with m = n, f = u, U = E; hence, the strategy u(t) = x'(t) (a.e.) is determined by the trajectory (a.e.). If A and B are as usual closed sets, then M = A x E is also closed, and f (t,x,u) is a given continuous scalar function on M. Here a function x(t) = (xl,...,xn), tl < t < t2, is an admissible trajectory provided x is AC in [tl,t2], (t,x(t)) e A for all t e [tl,t2], (t1,x(t1),t2,x(t2)) E B. and f (t,x(t), x'(t)) is L-integrable in [tl,t2]. Then the cost functional is I[x] = ft2 f (t,x(t),x'(t))dt. (2.14.1) The corresponding sets Q(t,x) and Q(t,x) have been already discussed in (2.12) (d), and the concept of weak seminormality has been introduced there. Our general statement (2.10.ii) in conjunction with (2.12.v) yields: (2.14.i). (A theorem of lower semicontinuity for free problems). For free problems (m = n, f = u, U = E ), if A is closed, if f (t,x,u) is continuous on M = A x E, convex in u, and weakly seminormal with respect to u in A, then the functional (2.14.1) has the property of lower semicontinuity; that is, if x(t) = (xl,..,x ), t < < t t, is an AC function which is the limit in the p-metric of admissible trajectories xk(t), tlk < t < t2k, k = 1,2,..., with p(xk,x) - 0, and lim I[xk] < + oo as k + + o, then x is admissible, and I[x] < lim I[xk]. The condition of weak seminormality is certainly satisfied if f (t,x,u) - + oo as lul * + oo for every (t,x) e A. Theorem (2.14.i) is due to L. Tonelli [ 1] who proved it for f of class 52

C' in u. A proof under the present sole continuity hypotheses was given by L.Turner [ 1]. The lower semicontinuity theorem (2.14.i) is here a corollary of theorem (2.10.ii) for lower closure of general Lagrange problems. Statement (2.14. i) without the hypothesis of weak seminormality is not true, as the following simple example shows. Take n = 2, A = E3, f = yx' - xy', 0 x,y state variables. Then f is certainly convex in (x',y'), namely linear. Nevertheless, I = ft2 (yx' - xy')dt is not lower semicontinuous. Indeed, if tl we take C: x = 0, y = 0, 0 < t < 2n, and Ck: x = k 1sin k2t, y = k cos k2t, o < t < 2n, k = 1,2,..., then C, I[Ck] = -2n, k = 1,2,..., and I[C] = 0. An analogous example for n = 1 has been given by L. Tonelli [3 ]. Nevertheless, Tonelli proved that, for n = 1, and f continuous in t,x,x' with continuous first order partial derivatives, statement (2.14.i) holds without weak seminormality requirement [1 ]. Again, the example above shows that this is not the case for n > 2. Remark 1. Generalized solutions for free problems form no TDS systems since the strategy (p,v) is not determined by the strategy as mentioned in (1.9). However, our general closure theorems (2.10.ii) and (2.10.ii) apply, and we state below, as an example, a corollary of (2.10.i) for generalized solutions and free problems. Here A is as usual a subset of the tx-space El+, U = E, M = A x E, we assume A closed, hence M is closed, and we assume f (t,x,u) continuous on M. A generalized solution is as usual (see (1.9)) a system x(t), p(t),(t)t), tl < t < t2, with x(t) AC and p(t), v(t) measurable in [tl,t2], (t,x(t)) E A for all t E [tl,t2], p(t) = (Pl,,Py), pj(t) > O, Zjpj(t) = 1 (that is, p(t) E r), v(t) = (u(,j = 1,...,), u (t) a.e. in 53

[tl,t2], satisfying dx/dt = Z71 pj(t)u(j)(t) a.e. in [tl,t2], and such that Z. pj(t)f(t,x(t),u(j)(t)) is L-integrable in [tl,t2]. Then the corresponding functional is I[xpv] = ft2 =1 pj(t)f (t'x(t),u(j)(t))dt, (2.14.2) (cfr. (1.9.3) and (1.9.4). Note that the sets R(t,x)c Ec +1 of all z = (z,z) = (z,z,...,z ) E with n+- n+l z>. pj f (t,xu(j)) z = p u(j)for some (p,v) E r x E, are exactly 330 33ny are exactln the sets co Q(t,x) if y = n + 2. We take for y the minimum integer for which this holds for all (t,x) E A, 1 < y < n + 2. (214. ii). (A theorem of lower closure for free problems and generalized solutions). For free problems (m = n, f = u, U = E ) and f (t,x,u) continuous on M = A x E, if the convex sets R(t,x) are closed and satisfy property (Q) at every point (t,x) E A with exception perhaps of a set of points whose t coordinate lies on a set of measure zero on the t-axis, let us assume that (*) for some locally integrable scalar function 4(t) we have f (t,x,u) > Y(t) for all (t,x,u) E M, with exception perhaps of another set of points whose t coordinate lies on a set of measure zero on the t-axis. Then the functional (2.14.2) has the property of lower closure; that is, if x(t) = (xl,...,xn), tl < t < t2, is any AC function which is the limit in the p-metric of the trajectories xk of generalized admissible systems xk(t),P(t), v(t), tk( < t < tk k4<t < tk12k' 5I41

k = 1,2,..., with p(xk,x) + O as k - oo, and lim I[xk,Pk,vk] < + +, then x is a generalized admissible trajectory, that is, there are measurable functions p(t) = (P,...,p ), v(t) = (u( ),j =,...,), t1 < t < t2, with p(t) E r, (J) u (t) E E, such that x(t), p(t), v(t), tl < t < t2, is a generalized admissible system, and I[x,p,v] < lim I[xk,pk,vk] as k -* a. A corollary of (2.10.i). Note that if Jul is of slower growth than f (t,x,u) as lul -+ o in A (or in some compact neighborhood A of the graph of x) then certainly all sets R(t,x) are closed and satisfy condition (Q) in A ( or in A ), and condition Ir is satisfied (with ir = constant in A ), as requested in (2.14.ii). This is a consequence of (2.12,Remark 1). 2.15. THEOREMS OF LOWER SEMICONTINUITY FOR PROBLEMS DEPENDING ON HIGHER DERIVATIVES Let us consider here problems concerning a functional of the form I[y] = t f(t,y(t),y'(t),..,y(n) (t))dt,(215.1) where y denotes a scalar function of t, where f is a given function of its n + 2 arguments, where we assume constraints of the form (t,y(t),...,y(n-1)(t)) E A c E+ boundary conditions of the form (n-l) (n-1) (t1,y(t1),...,y( )(tl),t2,y(t2),...,y (t2)) E Bc E2n+2 and where A and B are given subsets of the indicated spaces. Thus, no con55

straint on the values of y (n)(t). 1 2 =? n =(n-i) (n)and the By the substitution xl y, x2 = y.. x uand the use of the vector notation x = (x1,...,xn), the problem above is reduced to a Lagrange problem for the given n > 1, with m = 1, functional I[x] = t f (t,x(t),u(t))dt, (2.15.2) tJ 0 differential system dxl/dt n-l n dxl/dt = x2. dx /dt = x, dx /dt = u, (2.15.3) constraints (t,x(t)) E A, and control space U = El. As mentioned in (1.7) this is a TDS system, that is,the trajectory x determines the strategy u (by means of (2.15.3). This is the reason we have written I[x] instead of the customary I[x,u] in the first member of (2.15.2). Both statements (2.10.i) and (2.10.ii) yield corollaries for the situation above. We state here a corollary of (2.10.ii) which is rather general. Note that A is a subset of the tx-space E+ U = E, M = A x El, B a subset of the tlxlt2x2-space E2n+2' and f (t,x,u) is defined on M. For any 0 (t,x) c A we denote as usual by Q(t,x) the set of all z = (z~,z) E E2 with z > f (t,x,u), z = u, u E E1. We see that these sets Q(t,y) c E2 are the same as those we would consider in an analogous "free problem" concerning f (t,x,u) (though here n > 1, m = 1). We can speak of the seminormality and weak seminormality of f (t,x,u) with respect to u for u E E1 and (t,x) E A, as we did for free problems.

An admissible trajectory x(t) = (x',..,xn), t, < t < t2, is now any vector function satisfying the following requirements: (a) x1,...,xn are AC in [tlt2] and dxl/dt =2 n-l in [tl,t2] and dxl/dt = x2,...,dx /dt = xn for all t E [tl,t2] (thus u(t) = dxn /dt, or equivalently u(t) = dyn/dtn,y = x1, is certainly measurable and L-integrable in [tl,t2]; (b) (t,x(t)) e A for all t e [tl,t2]; (c) (tl,x(tl),t2,x(t2)) e B; (d) f (t,x(t),u(t)) is L-integrable in [tl,t2]. The corresponding functional is low (2.15.2). Of course, we shall use the p-metric on the vector functions x(t), tl < t < t2 (that is, on y(t), y'(t),...,y (n-1)(t) in the old notation). (2.15.i). (A theorem of lower semicontinuity for problems (2.15.1)). If A c E1+n is closed, if f0(t,x,u) is continuous on M = A x El, convex in u, and weakly seminormal with respect to u in A, then the functional (2.15.2) has the property of lower semicontinuity; that is, if x(t) = (x,,,,,xn), tl < t < t2, is an AC function, satisfying (a) above, is the limit in the pmetric of admissible trajectories xk(t), tlk < t < t2k, k = 1,2,..., with P(xk,x) O0 and lim I[xk] < + oo as k - o, then x is admissible, and I[x] < lim I[xk]. The condition of weak seminormality is certainly satisfied if f (t,x,u) + + oo as lul + + oo for every (t,x) E A. As a particular case of (2.15.1) we may consider a functional of the form I[y] = jt2 F(t,y(t),(Ly)(t))dt, (2.15.4) tl where L is a linear differential operator of the form 57

(Ly)(t) y(n)(t) + al(t)y (n-1)(t) +...+ a (t)y n If we take x = (xl,...,xn), l = y. = y(n-l) and f (t,x,u) = F(t,xl,u + al(t)xn +...+ a (t)xl) we have the same situation as above. Essentially the same theorem (2.15.1) holds where now we may require weak seminormality and convexity of F with respect to its third argument. The condition of weak seminormality is certainly satisfied if F is convex in u, and F(t,y,u) + + oo as Iul * + o for all (t,y). A great many possible extensions of problems (2.15.1) and (2.15.4) can be left as exercises for the reader. 58

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