T HE U N I VE R S I TY OF M I C H I GA. N COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Progress Report No 17 EXISTENCE THEOREMS FOR WEAK AND USUAL OPTIMAL SOLUTIONS IN LAGRANGE PROBLEMS WITH UNILATERAL CONSTRAINTS I. Closure Theorems Lamberto Cesari,OR!A; *,ioj~.'",.0..,'2';'.:;.:, -e.',.'G,".: uinder c.ntract'. wi t;i t NATIONAL SCIENCE FOUNDATION GRANT NO. GP-3920 WASHINGTON D. C administered through~ OFFICE OF RESEARCH ADMINISTRATION ANIN ARBOR September 1965

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EXISTENCE THEOREMS FOR WEAK AND USUAL OPTIMAL SOLUTIONS IN LAGRANGE PROBLEMS WITH UNILATERAL CONSTRAINTS I. CLOSURE THEOREMS* Lamberto Cesari Department of Mathematics The University of Michigan Ann Arbor, Michigan In the present papers (I, II, and III) 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] = fo(t,x(t),u(t))dt, tJ with side conditions represented by a differential system dx/dt = f(t,x(t),u(t)), t1 < t < t2, constraints (tyx(t))CA, u(t)CU(t,x(t)), t_ < t < t2, and boundary conditions (t1,x(t1),t2,x(t2) )eB, *Research partially supported by NSF-grant GP-3920 at The University of Michigan. 1

where A is a given closed subset of the tx-space El x En, where B is a given closed subset of the tlxlt2x2-space E2n+2, and where U(tx) denotes a given closed variable subset of the u-space Em depending on time t and space x. Here A may coincide with the whole space ElxEm, and U may be fixed and coincide with the whole space Em. In the particular situation, where the space U is compact for every (t,x) these problems reduce to Pontryagin problems; in the particular situation where the space U is fixed and coincides with the whole space Em, then these problems have essentially the same generality of usual Lagrange problems. Throughout these papers we shall assume U(t,x) to be any closed subset of Em. In paper I we prove closure theorems for usual solutions. In II we shall prove existence theorems. These will 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 NagumoTonelli existence theorem for free problems (m = n, f = u). In III we shall prove existence theorems for weak (or generalized) solutions introduced as measurable probability distributions of usual solutions (Gamkrelidze chattering states). In successive papers we shall extend the present results to multidimensional Lagrange problems involving partial differential equations in Sobolev's spaces with unilateral constraints. 2

We begin with an analysis of the concept of upper semicontinuity of variable subsets in Em. The usual concept of upper semicontinuity is replaced by two others (properties (U) and (Q), p3), which are essentially more general than the uppersemicontinuity, in the sense that closed sets U(t,x), for which uppersemicontinuity property hold, certainly satisfy j: property (U), and closed and convex sets Q(t,x), for which upper semicontinuity property hold, certainly satisfy property (Q). We then extend (X4) the closure theorem of A. F. Filippov in various ways, so as to include, among other things, the use of pointvise and not necessarily uniform convergence of some components of a sequence of trajectories. In part II we shall prove existence theorem of optimal smooth solutions (97) by a new analysis of a minimizing sequence, and by using the above extensiorf:of Filippov's closure theorem as a replacement for.o:_ie:lU''$ semicontinuity argument. We shall then deduce (~8) existence theorems for the case where f is linear in u, and for free problems of the calculus of variations (m = n, f = u). Finally, we shall prove (99) existence theorems for weak solutions in the general case above, for the case in which f is linear, and for free problems. 1. THE PROBLEM We denote by x a variable n-vector x = (x1,...,xn) En, by u a variable m-vector u (ul,...,um)e Em, and by teE1 the independent variable. We denote by A an arbitrary subset of the (t,x)-space, AcElxEn, and, for any (t,x)e A, we denote by U = U(t,x) a variable suspace of the u-space, U(t,x) c Em. In the terminology of control problems, u is the control variable 15

and U(t,x) the control space. We denote by fi(t,xu), i = Ol,1...,i1 given real functions defined for all (t,x)EA, and all ueU(t,x), and by f the nvector function f = (fi,.-,fn)- We denote by B a given subset of the (2.n + 2)-space (tl,xl,t2,x2). We are interested in the determination of a measurable vector function u(t), tl < t < t2, (control function, or steering function, or strategy), and a corresponding absolute continuous vector function x-(t), tl < t < t2, satisfying almost everywhere the differential system dx/dt = f(t,x(t),u(t)), tI < t < t2, satisfying the boundary conditions (t1,x(tl),t2,x(t2) )cB, satisfying the constraints (tx(t))cA, t t t2, u(t)eU(tx(t)), ae. in [t1,t2], and for which the integral (cost functional) t2 I[x,ul] fo(t,x(t),u(t) )dt tJ has its minimum value (see l 2 for details)~ We shall assume that U(t,x) is closed for every (t,x)EA. 2. THE SPACE OF CONTINUOUS VECTOR FUNCTIONS Let X be the collection of all continuous nadim. vector functions x(t) dei.:- on arbitrary finite intervals of the t-axis: x(t) = (x,O,,,xn), a < t K b, x(t)EEn,

If x(t), a < t < b, and y(t), c < t < d, are any two elements of X, we..shall define a distance p(x,y). First, let us extend x(t) and y(t) outside their intervals of definition by constancy and continuity in (- 0o, + o), and then let p(x,y) la-cl + lb-dl + max jx(t)-y(t), where max is taken in ( co, + oo). It is known that X is a complete metric space when equipped with the metric p. Ascoli's theorem can now be expressed by saying that any sequence of equicontinuous vector functions xn of X, whose graphs in the txOtsace are equibounded, possesses at least one subsequence which is convergent in the p-metric toward an element x of X. 35 ADMISSIBLE PAIRS u(t), x(t) Let A. be'a closed subset of the (t,x)-space ElxEn. For every (t,x)eA let U(tx), or control space, be a subset of the u-space Em. Let M be the set of all (Ut'x: u) with (t,x)eA,ucU(t,x). Let.'f(t,x,u) = (fl,...,fn) be a continuous vector function defined on M. We shall denote by Q(t,x) the set of all values in En taken by f(t,x,u) when u describes U(t,x), or Q(t,x) f(t,x,U(t,x)). A vector function u(t) = (u,...,), t < t < t2 (control n function) and a vector function (x,t) = (xl,...,x ), tl < t < t2 (trajectory) are said to be an admissible pair provided (a) u(t) is measurable in [tl,t2]; (b) x(t) is absolutely continuous (AC) in [t1,t2], (c) (t,x(t))eA for every t e[tl,t2]; (d) u(t)cU(t,x(t)) a.e. in [t1,t2]; (e) dx/dt = f(t,x(t), u(t)) a.e. in [tl,t2]. By the expression the vector function x(t), t., t < t2, is a trajectory, we shall mean below that there exists a vector function 5

u(t), t l < t < t2, such that the pair u(t), x(t) satisfies (abcde)o We say also that x(t) is generates by u(t). 4. UPPER SEMICONTINUITY OF VARIABLE SETS In view of using sets U(t,x),Q(t,x) which are closed but not necessarily compact, we need a concept of upper semicontinuity which is essentially more general than the usual one. We shall introduce two modifications of the usual definition of upper semicontinuity, and we shall denote them as "property (U)" and "property (Q)" since we shall usually use them for the sets U(t,x) and Q(t,x) above, respectively. We shall discuss properties (U) and (Q) first in relation to arbitrary variable sets U(t,x), Q(t,x) which are functions of (t,x) in A. Then we shall discuss.their relations when 4(t,x) is assumed to be the image of U(t,x) as mentioned in no. 2. Properties proved for U(t,x) under conditions (U) or (Q), will be used for Q(t,x) when this set satisfies conditions (U) or k) (A) -The Property (U) Given any set F in a linear space E we shall denote by cl F, coF, bdF, int F respectively the closure of F, the convex hull of F, the boundary of F, the set of all interior points of F. Thus, cl co F denotes the closure of the convex hull of F. We know that F, cl F, co F, co cl F are all contained in c.l co F. For every (tx)iA and t > Olet N(t,x) denote the closed oAneighborhood of (t,x) in A, that is, the set of all (t',x')EA at a distanced b from (t,x). 6

A. variable subset U(t,x), (t,x)cA., is said to be an upper semicontinuous function of (t,x) at the point (t,x)eA provided, given e > 0, there is a number 5 = b(t,x,C) > 0 such that (t,x) e N,(t,x) implies U(t,x) c [U(t,x)-] where [U]e denotes the closed e-neighborhood of U in Em. Again let U(t,x), (t,x)eA, U(t,x)CEm, be a variable subset of Em, which is a function of (t,x) in A. For every 5 > 0 let U(t,x,5) = UU(t',x'), where the union is taken for all (t',x') < Nb(t,x). We shall say that U(t,x) hasthe property (U) at (t,x) in A, if U(t,x) = [ cl U(t,x,). 5>0 We shall say that U(t,x) has property (U) in A, if U(t,x) has property (U) at every (t,x) of A. (i) If U(t,x) has property (U) at (t,x), then U(t,x) is closed. Indeed, U(t,x) C cl U(t,x)C A cl U(t,x, ) = U(t,x), 5>O and hence the c signs can be replaced by = signs. (ii) If A is closed, and U(t,x) is any variable set which is a function of (t,x) in A. and has property (U) in A, then the set of all (t,xu)eAxEm with ueU(t,x), (t,x)cA, is closed. Proof. If (t,x,u)ecl M and c > 0, then there are oo-many points (t,x,u)eM with It-ti < E, Ix-xl < c, < u-u < C. Thus, (t,x)eA. since A is closed, (t,x)cN2e(t,x), u U(t,x),uCU(t,x,2c), and ucf1cl U(t,x,2e) = U(t,x), ueU(t,x), since U has property (U) at (t,x). This proves that (F,,U)CM, that is, M is closed. 7

Note that the sets U(t,x,5) are not necessarily closed even if A is closed all sets U(t,x) are closed, and we take for N6(7f,Fx) the closed 6neighborhood if (t,x) in A as stated. This can be seen by the following example. Let A = [O < t < 1, O < x < 1] a subset of E2, and U(t,x) = Ez = (Z1i,z) iz2 > tzl,- 0< Z1< + *] for 0 < t < 1, and U(O,x) = [z2> 0, zl = 0] for t = 0. Then U(O,x,6) = [z = (zl,z2)I z2 > zz for -X< zj < 0, and z2 > 0 for 0 < zl< X ] for any 6 > O. The sets U(O,x,5) are not closed. Here U(t,x)' does not satisfy property (U) at the points (O,x). Nevertheless, the statement holds (iii) If A is closed, and U(t,x) satisfies property (U) in A, then the sets U(t,x,b), (t,x)eA, 6 > 0, are all closed, and hence U(t,x) = [!E U(t,x,6) for every (t,x)rA. Proof. Let MN denote the set of all points (t,x,u) with (t,x)e N6(t,x), ueU(t,x). Obviously N6(tx)c A c En+l; MEn+lxEm, and N6,(t,x)- is compact and M6 is closed by force of (ii) above. Let u be a point of accumnulation of U(t,x,5), and for any 0 > O let VRj(u) denote the ~-neighborhood of U in Em. Then M n(Vr(u)xEn+1) c:N,(t,x) xV~(u), hence M n(V(u)xEn+l) is bounded. Since both NM and VTl(.)xEn+l are closed sets, the set Mn(Vrl(u) xEn+l) is closed and bounded, and therefore a compact subset of En+l x Em. Now the set U(tx,5) n V.(U) is the projection of M [1(V~r(u)xEn+l) on the uspace Em, and therefore U(t,x,6)f lV(u) is compact. Thus UeU(t,x,5 )IV (u), and finally uc U(t,x,5) Thus, U(t,x,5) is closed, orcl U(t,x,5) = U(7,x,65), and U(U,x) = nf cl U(t,x,56) = ntu(t,x,5). 8

(iv) If A is closed and Uj(t,x), (t,x)eA, j 1,...,v, v finite, are — variable subsets of Em all-satisfying property (U) in-A, then their union " and their intersections V(tx) = UjUl(t,x), W(t,x) = njUl(t,x), (t, x)A, are subsets of Em satisfying property (U) in A. The same holds for their product V(t,x) = U'1 x...x U. The proof is straightforward. Under the hypotheses of (ii) the set M is closed but not necessarily compact as the trivial example U(t,x) = Em, M = AxEm, shows. The set M is closed but not necessarily compact even if we assume that A is compact, and that every U(t,x) is compact. This is proved by the following example. Let m = n = 1, A = [(t,x)6E2, 0 < t < 1, O < x < 1], U(O,x) = [ueE11O < u < 1], and, if t f O, U(t,x) = [ucEO10 < u < 1, and u = t-']. Then M is the set of all (t,x,u) with 0 < t < 1, 0 < x < 1, and 0 < u < 1, or u = t-1 if t Z O. Obviously, M is closed but not compact. Nevertheless, the statement holds: (v) If A is compact, if the variable set U(t,x) is compact and convex for every (t,x)eA and possesses property (U) in A, if for every (t,x)eA there is some 6 = b(t,x) > 0 such that U(t,x),iU(t',x') for every (t',x') eNF (t,x), then M is compact. Proof. If M is not compact, then there is some sequence of elements (tk,xkuk)e1M, k with (tk,xk)eA, Itk Ixk uk +. + Since A is compact and hence bounded, we have lukl [ + c. On the other hand, there is some subsequence, say still (tk,xk), with tk + t, xk + x, (t,x)EA. Given e > 0, we have ukEU(t,x,e) for all k sufficiently large, as well as U(t,x)llU(tk,xk) f p. Since U(t,x) is compact, there is a solid sphere S containing all of U(t,x) in its interior, say U(t,x) int S cEm. On the 9

other hand, if ukCU(t,x)fHU(tk,xk), we have uk e int S, and ukCEm-S, again for k large. Since both uk and uk belong to the convex set U(tk,xk), the segment ukuk is contained in U(tk,xk). In particular, if uj is the point where the segment ~uk intersects bd S, we have LTeU(tkxk) u eU(t,x,E), and utebd S. If u' is any point of accumulation of [uk], then u'ebd S, and u'W k cl U(t,x~,) for ever e > O. Hence, u'enecl U(T,x,) = U(t,x), a contradiction, since U(t,x)c int S. We have proved that M is compact. (vi) If the set U(t,x) is closed for every (t,x)cA and is an uppersemicontinuous function of (t,x) in A, then U(t,x) has property (U) in A. Proof. By hypothesis U(t,x,6)c [U(tx)]e, where Ue is closed. Hence cl U(t,x,5)c [U(t,x)] for 6 = 5(t,x,e) and any C > O. Since U(t,x) is closed, then [U(t,x)] U(t,x) as e - 0+. Thus nb5l U(t,x,5)c U(t,x). Since the opposite inclusion is trivial, we have nfcl U(t,x,5) = U(t,x). Statement (vi) is thereby proved. The uppersemicontinuity property implies property (A), but the converse is not true, that is the uppersemicontinuity property for closed sets is more restrictive than the property (U). This is shown by the following example in which all sets are closed. Take n 2 and 1 2 2 U(t,x) = [(u,u2)cE2l0u1 < + O < 2 < t] for every (t,x)eA = [(t,x)eE2O < t < 1, 0 < x < 1]. Then, for 5 > O, we have U(t,x,b) = [(u,u2)eE2I0 < u < + 0, 0 < u2 < (t+ )u1], hence U(t,x) = nscl U(t,x,5) and U(t,x) has property (U) in A. On the other hand, 10

[U(t,x)] = (u,u2 )E210 < u < + oo -e < u2 < tu +E(l+t 2 1/ 2 where N1 = NC (0,0) if t = 0, and, if t f O, N1 = N(00)((OO)L[,u )cuE21u <0, u u t >-t, _tul+U2 < e(l+t2)1/2]. Obviously U(t',x') - [U(t,x)]e 1 0 for t' > t, hence U(t,x) is not an uppersemicontinuous function of (t,x). (vii) If A is compact, if U(t,x) is compact for every (t,x)eA and is an upper semicontinuous function of (t,x) in A, then M is compact. (yiii) If A is closed and Uj(t,x), (t,x)eA, j = 1,...,v, v finite, are variable subsets of Em all uppersemicontinuous functions of (t,x) in A, then their union V(t,x) and their intersections W(t,x) are semicontinuous functions of (t,x) in A. The same holds for their product V(t,x) = U1 x...x Uv, as well as for their convex hull Z(t,x), that is, for the set Z(t,x) of all u = plul +...+ pvuv with uj e Uj(t,x), pj > O, j = l,...,v, Pi +...+P~ = 1. The proof is straightforward. (B) The Property (Q) Let U(t,x), (t,x)eA, U(tx)eEm, be any variable subset of Em, which is a function of (t,x) in A. By using the same notations as in (A), we shall say that U(t,x) has property (Q) at (t,x) in A., if U(t,x) n cl co U(t,x,6). 5>0 We shall say that U(t,x) has property (Q) in A if U(t,x) has property (Q) at every (t,x) of A. (ix) Property (Q) at some (t,x) implies property (U) at the same (t,x), and U(~,x) = nf cl co U(,xE,6) = n5 clU(~,x,~) = =,U(~,x,B). 11

Indeed U(t,x)C n cl U(T,x)e n cl U(txG)e n cl co U(,),Yj, 6>0 5>0 5>0 where first and last sets coincide by property (Q) at (t,x), and hence the inclusion signs c can be replaced by = signs. (xi) If A is closed, and U(t,x) is any variable set which is a function of (t,x) in A and has property (Q) in A, then the set M of all (t,x,u)eAxEm with uCU(t,x), (t,x)cA, is closed. Under the hypothesis of (i) the set M is closed but not necessarily compact as the trivial example U(t,x) = Em, M = AxEm shows. Nevertheless, the statement holds: (xii) If A. is compact, if the set U(t,x) is compact for every (t,x)eA and possesses property (Q) in A, then the set M is compact. Proof. If M is not compact, then there is some sequence, (tk,Xk,U k)eM, k = 1,2,..., with (tk,xk)eA, Itkl + IxkJ + lukl + + C as k oa. Since A is compact and hence bounded, we have lukl + + o. On the other hand, there is some subsequence, say still (tk,xk), with tk + t, xk + x, (t,x)eA. Given e > 0, we have then uksU(t,x,e) for all k sufficiently large. Since U(tx) is compact, there is a solid sphere S containing all of U(F,x) in its interior, say U(t,x)c int S c Em. On the other hand, if ueU(t,x), we have ue int S, and ukeEm-S, again for k large. Since both u and uk belong to the 12

convex set cl co U(t,x,e), we have U'ke cl co U(t,x,e) where uk is the point of intersection of the segment uuk with the boundary bd S of S. If u' is any point of accumulation of [u], then u'Ebd S, and u's cl co U(t,x,e) for every C > O. Hence u't nF cl co U(t,x,e) = U(t,x), a contradiction, since U(t,x)c int S. We have proved that M is compact. (xiii) If for every (t,x)eA the set U(t,x) is closed and convex, and U(t,x) is an uppersemicontinuous function of (t,x) in A, then U(t,x) has property (Q) in A. Proof. By hypothesis U(t,x,5)C [U(t,x)]E, where Uc is closed and convex as the closed c-neighborhood of a closed convex set. Hence, n5 cl co U(t,x,5)C [U(t,x)], for every e > O. Since U(t,x) is closed, then [U(t,x)]6 + U(t,x) as e - 0+. Thus n% cl co U(t,x,5)c U(t,x). Since the opposite inclusion relation D is trivial, we have nb cl co U(t,x,6) = U(t,x). (C) Relations Between Properties of U(t,x) and of Q(t,x) Let us now consider sets Q(t,x) = f(t,x,U(t,x)), (t,x)PA, Q(t,x)C En, which are the images of sets U(t,x)cEm for every (t,x)eA. The hypothesis that A is compact, that f is continuous on M, that U(t,x) has property (Q) [or (U)] in A, and that Q(t,x) is convex for every (t,x)eA, does not imply that Q(t,x) has property (Q) [or U] in A. This can be proved by a simple example. Let m = n = 1, A = [-1 < t < 1, 0 < x < 1], let U(t,x) be the fixed interval U = [ueEl1O < u < +o], and f = (u + 1) - t. Then Q(t,x) = [zeEll-t < z < 1 - t], and, if -1+5 < t < 1-5, then cl co Q(t,x,6) = -[-t-b < z < i-t+5].

The intersection of all these sets for 5 > 0 is the closed set [zeEl1 -t < z < l-t] which is larger than Q(t,x), and thus Q has not property (Q) in A. Actually, Q(t,x) is not closed, and hence Q(t,x) has neither property (Q), nor property (U). Even the stronger hypothesis that A is compact, that f is continuous on M, that U(t,x) has property (Q) in A, and that Q(t,x) is compact and convex for every (tx)EA, does not imply that Q(t,x) has property (Q) in A. This can be proved by the following example. Let m = 1, n 1, A = [(t,X)cE2, O < t < 1, 0 < x < 1], U = U(tx) = LuEllO <u < + a], and f(t,x,u) tu exp (l-tu), (t,x,u)eAxU. For t = 0 we have f 0 O, hence Q(O,x) = [z=0]. For 0 < t < 1, we have Q(t,x) = [0 < z < 11]. All sets Q(t,x) are compact and convex, but Q(t,x) does not satisfy property () nor property (U) in A. (xiv) If A is closed and f continuous on M, if U(t,x) is an upper semicontinuous function of (t,x), then Q(t,x) possesses the same property, and also has property (U). If we know that Q(t,x) is convex, then Q(t,x) has also property (Q). Proof. Each set Q(t,x) is a compact subset of En as the continuous image of the compact set U(t,x). Let us prove that M is closed. Let (r,x,u) be a point of accumulation of M. Then there is a sequence (tk,xk,uk) of points of M with tk + t, Xk + x, uk + u, and (tk,xk)eA, ukeU(t,x ). Then (t,x)EA since A is closed, (tkXk)eN6(tr,) for all k sufficiently large, and ukcU(tkxk)c [U(r,x)]C. Thus, uE[U(t,x)]e for every e > 0, and hence ~iTU(t,x) since this set is compact. We have proved 14

that (~,x,)CM, and that M is closed. Let us prove that Q(t,x) is an uppersemicontinuous function of (t,x). Given (t,x)eA and e > 0, let 5 = 5(t,x,C) > 0 be the number relative to the definition of uppersemicontinuity of U(t,x), and let M' be the set of all (t',xt,u') with (t',') CN5(t,x), u'cU(t',x'), and M" be the set of all (t',x',u') with (t',x')ENs (t,x), u'E[U(t,x)]E. Since U(t,x) is compact, also [U(t,x)]e is compact. Hence M" = Nb(t,x)x[U(t,x)]E, and M' = MFM". The set M' is compact as the intersection of the closed set M with the compact cylinder M". The function f is continuous on M' and hence bounded and uniformly continuous. Hence, there is some i, 0 < n < min [6,1], such that (t",x")NT(tx')' Iu'-u"' <A,n, (tt,x'',u), (t",x",u")eM' implies If(t',x,u?)-f(t",u")l <. Also, let a = min [n, 6(tx,rj)]. Then, for every (t',x')eNa(t,x), we have U(t',x') c[U(t,x)], hence, if u'EU(t',x'), there is some u1"U(t,x) with lu'-u"I < A, and finally If(t',x),u')-f(t,x,u")i < C. Thus, Q(t',x')c [(Q(t,x)]E for every (t',xt)eN' (t,x). This proves that Q(t,x) has the e6-property above. The last part of statement (xiv) is now a consequence of statements (vi) and (xiii). Remark. The statements and examples above show that properties (U) and (Q) are generalizations of the concept of upper semicontinuity for closed, or closed and convex sets, respectively. (xv) If A is a closed subset of the tx-space F&xEn, if U(t,x), (t,x)eA, U(t,x)c Em} is a variable subset of Em satisfying property (U): in A, if M denotes the set of all (t,x,u) with (t,x);A, ucU(t,x), if fo is a continuous scalar function from M into the reals, if'(t,x) denotes the variable sub15

set of Eml+ defined by U(t,x) [u = (u,u)cEm+lU~0 > fo(t,x,u), ucU(t,x)], then Ui(t,x) satisfies property (U). Proof. First, let us prove that each set U(to,xo,5) is closed. Indeed, if u = (u~,u) is a point of accumulation of U(to,xo,5), then there is a'0'' 0 0 sequence uk (uk,uk) with uk + u, uk + u, ukEU(to,xo,5). Hence, there is a corresponding sequence of points (tk,xk)eN6(to,xo) with uk > fo(tk,Xk,Uk), ukeU(tk,Xk). Thus ukEU(to,xo,5). Since Nsj(to,xo) is a compact part of the closed set A, there is a subsequence, say still (tk,xk), with tk + t, Xk + x, (t,x)N(toxo)CA. Thus (tk,xkuk)CM, (tk,xky)+k) + (t-,xu), and M is a closed set by force of (ii). By the continuity of fo we have then (t,x,u)-M, u:(tx),: uo > fo:(tx,u). Thus u= (uO,u)e~U(t,x), and ueU(to, 0 Now let u (u,u) be a point ucs fl cl U(to,xo,q ). Thus, there is a sequence of numbers k > O, k O, with ue cl ~(to,Xo,%k), and hence uitU(to,xo,Sk) because these last sets are closed. Thus, there is also a sequence of points (tk,xk)N(to,xo) with uU(tk,xk), or u~ > fo(tk ) uCU(tk,xk). Hence, for every T > O, we have ucU(to,xo,A) for every k sufficiently large (so that Sk q ), and, by property (U) of U(t,x) at (t0,xo), also us f1 cl U(to,xoy,) = U(to,xo). Thus, uUJ(to,xo), (to,Xo,U)eM, and by O 0 u > fo(tkxku) and the continuity of fo, also u > fo(to,xo,u). We have proved that u - (u,u)U(to,xo), hence n5 cl U(toXo, )c U.(t,Xo ). Since the opposite inclusion relation is trivial, equality sign holds, and J(t,x) has property (U) at (to,xo), and, thus, everywhere in A. Statement 16

(xv) is thereby proved. The set U(t,x) of statement (xv) has not necessarily property (Q) even if we assume that U(t,x) has property Q and fO(t,x,u) is convex in u for every (t,x)eA. This can be seen by a simple example. Let A = [-1 < t < 1, 0 <x < L] and let U = U(t,x) be the fixed set U(t,x) El, that is, U = [- 0 < ul < + co]. Then, each set U(t,x) is closed and convex, and obviously U(t,x) possesses property (Q), and M is the cylinder of all (t,x,u) with (t,x)eA, ucEl. Finally, let fo(t,x,u) = tu', so that fo is continuous in M and, for every (tx)eA, fo = tul is linear in u1, hence 1 certainly convex in u. Now we have J(t,x) = [(uO,Ul)E2l oo < u < +, tu < < + o] ~U(O XI), -[(u,ul)cE21- < U < +, b j1uij< u < + 0. Consequently, co U(O,x,,) - E2, and hence n cl co U(0,x,6) E2, while U(Ox) (Iu2)E2j. co < U1 < + co U2 > 0. This shows that U(t,x) does not have property (Q) at the points (O,x) of A. A scalar function fo(t,xu), (tx,u)eM, is said to be convex in u at (to,Xo)eA if N fo (to,XO,u) _<. ro(to,xo,u), i-=l whenever uo = kiui * 2

where ui C U(toxo), ki > O, i = 1,..., N, xi + + AN = 1. A. scalar function fo(t,x,u), (t,x,u) + M, is said to be quasi normally convex in u at (to,xo,uo)eM provided, given E > 0, there are a number 8 = 6(to,xo:,uo,e) > 0, and a linear scalar function z(u) = z + b u, b = (b,...,bm), r, bl,..,bm real, such that (a) fo(t,x,u) > z(u) for all (t,x)eN5(to,xo),uyU(t,x), (b) fo(t,x,u) < z(u)+e for all (t,x)eN6(to,xo),ueU(t,x), lu-uo 1< 6. The scalar function fo(t,x,u) is said to be normally convex in u at (to,xo,uo) if, given c > 0. there are numbers 5 b6(to,xo,uoE) > 0, v = V(toxouo,) > 0, and a linear scalar function z(u) = r + bwu as above such that (b) holds and (a') fo(t,x,u) > z(u) + vIu-uol for all(t,x)eN6(to,xo),uCU(t,x). The scalar function fo(t,x,u) is said to be quasi normally convex in u, or normally convex in u, if it has these properties at every (to,xo,uo)eM. For the case where U = U(t,x) is the fixed set U = Em,, the following statement gives a useful characterization of the functions fo which are normally convex in u. (xvi) If A is closed, and fo(t,x,u) is continuous on M = AxEm, then fo is normally convex in u if and only if f is convex in u at every (to,Xo)EA, and for no points (to,xo)eA, uo,ulcEm, ul, 0, the relation holds fo(to,xo,,uo) = 2' [fo(to,xo,uo+2ul) + fo(toxo,uo-%ul) for all X > 0. This statement was proved in [9a] and [10]. In particular, if for every (t,x)eA, fo(t,x,u) is convex in u and fo(t,x,u)/lul - + co as lul. + + +, then certainly fo(t,x,u) is normally convex in u. 18

(xvii) If A is closed subset of the tx-space E1 x En, if U(t,x), (t,x)cA, U(t,x)cEm, is a variable subset of Em satisfying property:(Q) in A, if M denotes the set of all (t,x,u) with (t,x);A, ueU(t,x):, if fo is a continuous function from M into the reals, which is convex in u for every (tjx)eA, if either (a) the sets U(t,x) are all contained in a fixed solid sphere S of Em, or (B) the function fo(t,x,u) is quasi normally convex in u at every (to,xo,uo) of M, then the set U(t,x) of statement (xv) has property (Q) in A. Proof. Let u = (u~,u) be a point ~ = fn cl co U(to,xo,5). Then there is a sequence [Sk] of numbers k > 0, Sk + O, with Eucl co U(to,xo,Sk). Hence, there is a sequence of pairs of points Ukl, uk eEm+l and of points vk of the segment (uk ik2)~Em+lj such. that vk + u,: Ukluk2cU(to,Xobk) Vk 0k6kl + (lug ukkS, O < oi < 1, k = 1,2,.... We shall use the notation vk = (vIvk), u = (u~,u), ~kj = (U:ukj), j = 1,2. Then we have 0 O vk + u, k + u, uk1,uk2 CU(to,Xxo,k) C +,-k 1 (gllO (lo)uk, Vk =OIukl+ (u1 )Uk2 Consequently, there are points such that (tk,xk ), (tk2,Xk2)Nbkl(toXo) A, uklyU.(tkl1xkl),Uk2eU(tk2,xk2) 3,9

The sequence [ckk] is bounded, hence there is a convergent subsequence, say still ok, so that o0 + - for some 0 < a < 1. For every r > 0 and k sufficiently large (so that 5k < Tr), we have UklUk2cU(to,xoyr), hence Ukl,Uk2ECl CO U(to,Xon). As a consequence Vk -= Oxukl +(1'-)c k U Ec l co U(to,x, ) for all k sufficiently large. As k + oo, we obtain uccl co U(to,xo,0). By the property (U), finally uencll co U(t,x,) = U(t,Xo = U(oo) () Assume first that condition (5) holds. Then both sequences [ukl], [uk2] are bounded, and hence there is a subsequence, say still [ukl], [uk2], for which both uk1 and uk are convergent in Em, say ukl + ul, uk2 - u2, ul,u2cEm. For such a subsequence, we have vk = kUk + (1-k))Uk Ofo(tkl XUk) + (l%)fo(tk XkU )+ vk = Ouk1 + (l-oUk)Uk2, (tklXkUukl) (tk2,X2kauk2)aCM where M is closed. By taking limits as k -+ oo we have u > aofo(to,xo,ul) + (l-)fo(toxo,xo 2) u = a + (1-.O)u2, (to,Xo U1), (to,Xo',u2)EM. By the convexity of fo in u at (to,xo) we have now 20

u~ > fo(to,Xo,CM1 + (1-UC)u2) = fo(to,xo,u). This proves that u = (u~,u)eU(t0,xO), hence n.. cl co U(to,xo)c U(t,Xo) ( Since the opposite inclusion is trivial, = sign holds in this relation, and U(t,x) has property (Q) at (to,xo). Since (to,xo)eA is arbitrary, U(t,x); has property (Q) in A. Assume now that condition.(I) holds. As stated by relation (1) above, uCU(t6,XO), hence.(tO,xo,u)GM, By the quasi normal convexity of fo in u at (to,xo,u) we deduce the existence of a number 5 > 0 and of a linear scalar function z(u) = r +b. v-such i that (a) fo(t,x,v) = z(v) for all (t,x)ENs(to,xo), vEU(t,x) and (b) fo(t,x;v) < z(v) + e for all (t,x)eN6(to,xo), veU(t,x), lu-vI < 6. By combining (a) and (b) we have then (c) z(u) < fo(to,xo,u) = Z(U) + c. Now we have vk = OCukl +(l-ak)uko for some 0 < cO < 1, and vk + u (tkjxkj) + (to,xo), j = 1,2. Thus, for k sufficiently large, (tkj,xkj)eNN (to,Xo), j = 1,2, and, by property (a), _.-':'..-:._ -E Vk > OCkfo(tk,xklukl) + (l-Ck)fo(tk2Xk2 Uk2) > kZ(uk1) +.(l~-k)z(u k2) = z(cukx + (zl-c)Uk2),7 Z(Vk ) 2' As-:k++oo, we hav:~vec.t %e:tan~ u"_0> z'(u), and finally by'(c) above, u~ > fo(tOxou) -e,'w'here > 0 is arbitrary. We conclude that u~ > fo(to,xo,u),:wit: uCU(toxoX)' Thus U' (u,v)cU(to,xo), and'again we have proved inclusion (2). The same reasoning above yields that U(t,x) has property (Q) in A. 21

(xviii) If A is a closed subset of the tx-space Ei x En, if U(t,x),; (t,x)CA, is a variable subset of Em satisfying property (U) in A, if M denotes the set of all (t,x,u) with (t,x)eA, ueU(t,x), if f = (fo,f) is a continuous function from M into the z-space En, z (z,z), if Q(t,x)c En Q(t,x)c En+l are the sets Q(t,x) = f(t,x,U(t,x)) = [zEEnlz = f(t,x,u),ueU(t,x)3 - O ~(t,x) = [z = (zO,z)EEn+lIZO > fo(t'xu), z = f(tx,u), uEU(t,.x)], and (a) for every (t,x)CA, Q(t,x) is a convex subset of En; (b) Q(t,x) has propriety (Q) in A; (c) for every (t,x)EA, z = f(t,x,u) is a 1-1 map from U(t,x) onto Q(t,x) with a continuous inverse u = frl(t,x,z), zeQ(t,x); (d) the real valued function Fo(t,x,z) = fo(t,x,f-Z(t,x,z)), (t,x)eA, zQ(t,x), is continuous in the set M' of all (t x,z) with (tx)eA, zeQ(t,x), and Fo(t,x,z) is convex in z and also quasi normally convex, then the set Q(tx) is convex and has property (Q) in A. Proof. Indeed, under the specific hypotheses above, the set Q(t,x) can be represented as Q&~,(tx) =z > Fo(txz), zQ(t,x)], and thus Q is generated from Q(t,x) exactly as U is generated from U(t,x). By statement (xvii) above we conclude that Q(t,x) has property (Q) in A. Remark. The condition that f is a homeomorphism. between U and Q is certainly verified in all free problems, where m n, f = u, that is, fi u= i, i = 1,2,...,n (see no. 9 below). In this situation then we have Fo(t,x,u) = fo(t,x,u), and the convexity of fo in u implies the convexity of 22

Fo in u. We shall need this remark, and the more general statement (xviii) in part II. 5. CLOSURE THEOREMS We shall use here the notations of no. 2 and 3. In particular, a trajectory x(t) is defined as in no. 2. Closure Theorem I. (A first generalization of Filippov's theorem). Let A be a closed subset of E1 x En, let U(t,x) be a closed subset of Em for every (t x)eA, let f(t,x,u) = (fl,..,fn) be a continuous vector function on M into En, and let Q(t,x) = f(t,x,U(t,x)) be a closed convex subset of En for every (t,x)eA. Assume that U(t,x) has property (U) in A, and that Q(t,x) has property (Q) in A. Let xk(t), tlk < t < tk, k = 1,2,..., be a sequence of trajectories, which is convergent in the metrix p toward an absolutely continuous function x(t), t1 < t < t2. Then x(t) is a trajectory. Remark. If we assume that U(t,x) is compact for every (t,x)eA, and that U(t,x) is an uppersemicontinuous function of (t,x) in A, then by statement (x), the set Q(t,x) has the same property, U(t,x) has property (U), Q(t,x) has property (Q), and closure theorem I reduces to one of A. F. Filippov [2] (not explicitely stated in [2] but contained in the proof of his existence theorem for the Pontryagin problem with U(t,x) always compact). Proof of Closure Theorem I. The vector functions cp(t) = x'(t), tl < t < t2,() pk(t) = ~x(t) = f(t,xk(t),uk(t)), t < t < t k = 1,2,.., are defined almost everywhere and are L-integrable. We have to prove that 23

(t,x(t))eA' for every tf < t < t2, and that there is an admissible control function u(t), tl < t < t2, such that p(t) = x,(t) = f(t,x(t),u'(t)), u(t)cU(t,x(t)), (2) for almost all te[tlt2]. First, P(xk,X) 0+ as k 0; hence, tlk+ tl, t2k + t2. If te(tl,t2), or tl < t < t2, then t.k < t < t2k for all k sufficiently large and (t,xk(t))6A. Since xk(t) + x(t) as k + oo and A is closed, we conclude that (t,x(t))eA for every tl < t <t2. Since x(t) is continuous, and hence continuous at tl and t2, we conclude that (t,x(t))eA. for every tz < t < t2. For almost all te[tlt2] the derivative x'(t) exists and is finite. Let to be such a point with tl < to < t2. Then there is a a > O with tl < to - a < to + a < t2, and, for some ko and all k > ko, also tzk < to - a < to + a < ttk. Let xo = x(to). We have xk(t) + x(t) uniformly in [to - a, to + a] and all functions x(t), Xk(t) are continuous in the same interval. Thus, they are equicontinuous in [to - a,,to + ]. Given e > O, there is a 6 > 0 such that t,t'e[to -, to + a], It-t't i<, k > ko, implies Ix(t) - x(t')l < e/2 Ixk(t) - xk(t')l < ~/2. We can assume 0 < 6 < a, 5 < e. For any h, 0 < h < 5, let us consider the averages h mh = h-'fcp(to+s)ds = h (t-th)-x(tO)] h 0(3) mhk. = h-flctJ ( to+s)ds = h-l[xk(to,*h)-xk(to ) ] 24

Given rT > 0 arbitrary, we can fix h, 0 < h < 6 < c, so small that Imh- (to)l <. (4) Having so fixed h, let us take kl > ko so large that Imhk - mhlX(to) < xk(to)-X(to) < /2 (5) for all k > ki. This is possible since xk(t) + x(t) as k + o both at t =to and t = to + h. Finally, for 0< s < h, Ixk(to+s)-x(to) I < I xk(to+s)-Xk(to) +xk(to )-X(to) I < C/2 + c/2 -, I (to+s)-to < h < 6 < e f( to.,xk(t o +s),uk(to+s) )EQ(to+S,x(to+) ). Hence, by the definition of Q(to,xo,2E), also cpk(to+s) =f(to+s,Xk(to+S), uk(to+s))~Q(toxo 02o) ). The second integral relation (3) shows that we have also mhk cl co Q(to Xoj2e), since the latter is a closed convex set. Finally, by relations (4) and (5), we deduce i(p(to)-m _ < I(p(to)-mhl+lmh-mhk I < 2h, and hence cp(to)e [cl co Q(to,Xo,2e)1]2q. Here T > 0 is an arbitrary number, and the set in brackets is closed. Hence, cp(to)E o co Q(txo,,2c), 25

and this relation holds for every e > O. By property.(Q) we have. cP(to) e ne cl co Q(to,xo,2e) = Q(to,xo), where xo = x(to), and Q(to,Xo) = f(to,xo,U(toxo)). This relation implies that there are points u = u(to)eU(to,xo) such that (p(to) = f(to,X(t o),u(to) )) (6 This holds for almost all toC[ti,t2], that is, for all t of a measurable set I c [tl,t2] with meas I = t2-tl. If we take Io [tl,t2]-I, then means Io = O. Hence, there is at least one function u(t), defined almost everywhere in [tlt2], for which relation (6) holds a.e. in [tl,t2]. We have to prove that there is at least one such function which is measurable. For every teI, let P(t) denote the set P(t) = [ujueU(t,x(t)),cp(t) = f(tx(t),u] U(t,x(t))c Em. We have proved that P(t) is not empty. For every integer x = 1,2,...., there is a closed subset Ck of I, C> qC [tl,t2], with meas Ck > max [O,t2-tl-l/X], such that cp(t) is continuous on A. Let Wk be the set wA = [(t,u)IteCc, uCP(t)]c E1 x Em. Let us prove that the set W% is closed. Indeed, if (t,u) is a point of accumulation of W., then there is a sequence (ts,us),s = 1,2,..., with (ts,us)CeW, ts - E, Us + U. Then tseCC and tcCk since CA is closed. Also x(ts) + x(t), cp(ts) + cp(~), and since (ts, x(ts))eA, cp(ts) =f(ts,x(ts), u(ts)), (ts,x(ts), u(ts))EM, we have also (t,x(t))eA, (t,x(r),u)eM, because 26

A and M are closed, and cp(t) = f(,x(t),u) because f is conrtinuous.; Thus, uTP( ), and (t,x)eWh.; i For every integer R let Wj, PR(t), be the sets W., = [(t,u) (t,u)EWx, iuI _ < ] c WXcEi x Em, Pe(t) = [ulueP(t),IuI < _]CP(t) cU(t,x(t))cEm, C%. = [t (t,u)eWX2 for some u] c CAcIc[tlt2]. Obviously, Wg is compact, and so is CR as its projection on the t-axis. Also, U1C, = Cx, and Win is the set of all (t,u) with tECp, uscPy(t)-. Thus, for tcCx, P(t) is a compact subset of U(t,x(t). For teC.g, the set Py(t) is the nonempty compact subset of all u = (u',... u )eU(t,x(t)) with f(t,x(t),u) = p(t). As in Filippov's argument let P, be the subset of P with ul minimum, let P2 be the subset of P1 with u2 minimum,..., let Pm be the subset of Pm_l with um minimum. Then Pm is a single point u = u(t)eU(t,x(t)) with u(t) = (uJ,...,um), teCh,, Ju(t)l K<, and f(t,x(t),u(t)) =p(t). Let us prove that u(t), teChJ is measurable. We shall prove this by induction on the coordinates. Let us assume that ul(t)...*,uSl-(t) have been proved to be measurable on CkQ and let us prove that uS(t) is measurable. For s = 1 nothing is assumed, and the argument below proves that ul(t) is measurable. For every integer j there are closed subsets Cj of CQ with C C, Cj C C +l meas Ce > [0, meas: C -l/j], such that u1(t),...,u S(t) are continuous on Cj. The function cp(t ) is already continuous on C. and hence p(t) is continuous on every set Co and Ckj Let us prove that u (t) is measurable on Cxej. 27

We have only to prove that, for every real a, the set of all tEcCj with cps(t) < a is closed. Suppose that this is not the case. Then there is a sequence of points tkcCYjj with uS(tk) < a, tk + teChj, uS(t) > a. Then cp(tk) + (P(F), u (tk) + u (t) as k + a, cz = l,...,s-l. Since lu1(tk)I < 2 for all k and P = s,s+l,...,m, we can select a subsequence, say still [tk] such that u (tk)+u as k + o, X = s,s+l,..*,m, for some real numbers u. Then tk +t, x(tk) + x(t), u(tk) + u, where u = (ul(t)), U (t) (,u, ) Thern given any number T > O,we Eve U(tk )tU(tkX(tk) ) cl U(tx(t),d) for all k sufficiently large, and, as k + oo, also u e cl U(t,x(t),n). By the property (U) we have uE nl cl U(t,x(t),1) = U(tx()) On the other hand c(tk) = f(tk,x(tk),u(tk)), u (tk) < a, yield as k + 00, p(t>)= f(t,x(t) u), u <a (6) while tECXA implies cp(t) = f(t,x(t),u(t)), u (t) > a. (7) Relations (6) and (7) are contradictory because of the property of minimum with which uS(t) has been chosen. Thus uS(t) is measurable on CXh for every j, and then uS(t) is also measurable on CA. By induction argument, all components ui(t),...,um(t) of u(t) are measurable on Ce, 28

hence u(t) is measurable on C.. Since UgCk2 = CC, meass CN > measz I - 1/X, we conclude that u(t) is measurable on every set Ck and hence on I, with meas.- I = t2-t1. Thus, u(t) is defined a.e.. on (tl,ta), u(t)EU(t,x(t) and f(t,x(t),u(t)) = cp(t) a.e. on [tl,t2 ]. Closure Theorem I is thereby. proved. Let us denote by y = (xI,..,xs) the s-vector made up of certain comn ponents, say xl,.,X, O < s < n, of x = (xl,..,x ), and by z the pomplementary (n-s)-vector z = (xS,...,x ) of x, so that x = (y,z). Let s us assume that f(t,y,u) depends only on the coordinates x,...,x of x. If x(t), tl < t < t2, is any vector function, we shall denote by x(t) = [y(t),z(t)] the corresponding decomposition of x(t) in its coordinates y(t) = (xl,...,xS) and z(t) = (Xs+,...,n). We shall denote by A a closed subset of points (t,xl,...,xs), that is, a closed subset of the ty-space ElxEs, and let A - Ao x Ens. Thus, A is a closed subset of the tx-space ElxEn. Closure TheOrem II. (A further generalization of Filippov's Theorem). Let Aobe a closed subset of the ty-space E-xEs, and then A - Ao x Ens is a closed subset of the tx-space EzxEn. Let U(t,y) denote a closed subset of Em for every (t,y)eA,, let Mo be the set of all (t,y,u)eE +s+m with (t,y)cAo, ucU(t,y), and let f(t,y,u) = (f~z,... fn) be continuous vector-function from M into En. "Let Q(t,y) = f(t,y,U(t,y)) be a closed convex subset of En for every (t,y)eA0. Assume that U(t,y) has property (U) in Ao and that Q(t,y) has property (Q) in A0o 29

Let xk(t), tlk < t < t k k = 1,2,..., be a sequence of trajectories, xk(t) = (yk(t), zk(t)), for which we assume that the s-vector yk(t) converges in the p-metric toward an AC vector function y(t), tl < t < t2, and that the (n-k)-vector zk(t) converges (pointwise) for almost all tl < t < t2, toward a vector z(t) which admits of a decomposition z(t) = Z(t) + S(t) where Z(t) is an AC vector function in [tlt2], and S'(t) = O a.e. in [tl,t2] (that is, S(t) is a singular function). Then, the AC vector X(t) = [y(t),Z(t)], tl < t < t.2 is a trajectory.Remark. For s = n, this theorem reduces to closure theorem I. Proof of Closure Theorem II. The vector functions cp(t) = X'(t) = (y'(t),Z'(t)), tl < t t2, cp(t) = xi(t) = (y(t),zjt(t)) = f(t,yk(t),uk(t)), t k < t < t2k (8) k= 1,2,... are defined almost everywhere and are L-integrable. We have to prove that [t,y(t),Z(t)]eA for every tl < t < t2, and that there is an admissible cqntrol function u(t), tl < t < t2, such that cp(t) = X'(t) = (y(),Zt)) = f(t,y(t),u(t)), u(t )U(ty(t ) ), for almost all t c[tl,t2]. First, P(yk,y) + Oas k + O; hence tlk - tl,t9 -t t2. If te(tl,tp), or tl < t < t2, then tlk < t < tk for all k sufficiently large, and (t,yk(t)cAo. Since Yk(t) + y(t) as k ~+ o and A0 is closed, we conclude that (t,y(t))cA0 for every tl < t < t2, and finally (t,y(t),Z(t))cA.0 x Ens, 3o

or (t,x(t))cA, t1 < t < t,2 For almost all tc-[t1,t2] the derivative X'(t) - [y'(t),Z'(t)] exists and is finite, S'(t) exists and S' (t) = 0, and zk(t) + z(t). Let to be such a point with t1 < to < t2. Then there is a a > O with t1 < to - 0 < to + a < t2, and, for some ko and all k > ko, also t k < to a < to + a < t. Let xo = X(to) = (Yo,Zo), or yO = y(to), Zo = Z(to)' Let zo = z(to ), So 0 S(to). We have S'(to) = O, hence z'(to) exists and zt(to) = Z'(to). Also, we have Zk(to) z(to)We have Yk(t) + y(t) uniformly in [to - a, to + a], and all functions y(t), Yk(t) are continuous in the same interval. Thus, they are equicontinuous in [to - a, to + a]. Given c > 0, there is a 5 > 0 such that tt' e[to a-, t + a], It - t'l < 5, k > ko, implies IY(t) - y(tt)l < c/2, (t) - Yk(t' )! < e/2. We can assume 0 < 6 < a, 6 < C. For any h,O < h < 6, let us consider the averages h mh = h J (to s)ds h -L[X(to + h)- X(t))], o (10) mhk= h f (to + s)ds = h l[xk(to + h) Xk)], where X (-(y,Z), X = (YkXzk)k Given r > 0 arbitrary, we can fix h, 0 < h < < a so small that Imh - p(t0o) < I, IS(t0 + h) - S(to)f < rh/4, 31

This is possible since h-' cP(to + s)ds - cp(to) and [S(to + h) - S(to)]h 0 + 0 as h + 0 + o Also, we can choose h, in such a way that zk(to + h) + z(to) as k + + co This is possible since zk(t) + z(t) for almost all tl <t <t2o Having so fixed h, let us take kl > ko so large that Yk(to) - Y(to) I, Yk(to + h) - y(to+h) <I min [vh/4, c/2]:, Izk(to) - z(to)J, Izk(to + h) - Z(to+h)l < qh/8. This is possible since yk(t) + y(t), zk(t) + z(t) both at t = to and t - to + h. Then we have Ih Tyk(to + h) yk(to)] h [y(to + h) y(to)]l < fI-[Yk(to + h) y(to +-h)]l + Ih-[yk(to) y(to)]l < h-'(h/4) + h-(h/8) < /2. Analogously, since z = Z + S, we have Ih-l[zk(to + h) zk(to)] h — -Z(to + h) - Z(to)]l = Ih-[zk(to + h) - zk(to)] h-l[z(to + h) - z(to)] + h-l[S(to + h) - S(to)] I < Ih~-'[zk(to + h) z(to + h)] +lh-[zk(t) - (t o)]l+Jh'M[S(to + h) - S(to)]l < h l(h/8) + h- (~h/8) + h-1(rh/4) - r/2. Finally, we have Imk~ -- mhl i Ihl[xk(to + h) - xk(to)] - h'l[X(to + h) - X(to)]l IhL[Yk(to + h) - yk(t)] - hEl[y(ta + h) - y(to)]! +

+ Ih-l[zk(t + h) Zk(to)1 - h-L[Z(t + h) - Z(to)l < T/2 + T/2 = v. We conclude that for the chosen value of h, 0 < h < 5 <~ a, and every k > kl we have Imh - p(t0o)l < r, Imhk - mhl < r, IYk(to) Y(to)l _ /2 -(11) For 0 < s < h we have now lYk(to + s) - y(to)I < lyk(to + s) - Yk(to)l+lYk(to) - Y(to)l < K/2 + e/2 = I(to + s) - tol < h < < - K, f(to + syk(to + s),uk(tO + s))eQ(to + a,yk(to + s)). Hence, by definition of. Q(to,yo,2e), also Cpk(to + s) = f(to + S,yk(to + s), uk(to +s:))eQ(toYo,2)The second integral relation (10) shows that we have also mhk e cl co Q(to,Yo,2E) since the latter is a closed convex set. Finally, by relations (11), we deduce Icp(to) - mhkI < IcP(to) -mhllmh - mb < 2, and hence cp(to) e[cl CO Q(to,yo,2c)] 2T' Here T > O is an arbitrary number, and the set in brackets is closed. Hence P(to)E c1 co Q(toYo,2.), and this relation holds for every c > O. By property (Q) we have 33

cp(to)e nfcl co Q(to,yo,2c) = Q(toYo), where yo = y(to), and Q(to,Yo) = f(to,Yo,U(to,yo)). This relation implies that there are points u = a(to)eU(to,yo) such that cP(to) = f(toY(to),j(to)). This holds for almost all to e[t1,t2]. Hence, there is at least one function U(t), defined a.e. in [tl,t2] for which relation (9) holds a.e. in [tl,t%]. We have to prove that there is at least one such function which is measurable. The proof is exactly as the one for closure theorem I, where we write YYk instead of x,xk, and will not be repeated here. Closure theorem II is thereby proved. Content. 1. The problem. p. 5.-2. The space of continuous vector functions. p. 4 —53. Admissible pairs x(t), u(t). p. 5.-4. Uppersemicontinuity of variable sets. p. 6. —5. Closure theorems. p. 23. 34

REFERENCES 1. L. Cesari (a) Semicontinuitae convessita nel calcQlo delle varlazioni, Annali Scuola Normale Sup. Pisa, 14, 1964, 339-423.; (b) Un teorema di esistenza in problemi di controlli ottimi, Ibid., 1965, 35'-7. (c) An existence theorem in problems of optimal control, J. SIAM Control, to appear;: (d) Existenc'e Theorems in problems of Iagrange and optimal control, J. SIAM Control, to appear. 2. A.. F. Filippov, On certain questions in the theory of optimal control, Vestnik1 Moskov. Univ. Ser'. Mat. Mech. Astr. 2, 1959, 25-32 (Russian). English translation in J. SIAM Control, (A) 1, 1962, 76-34S. 3. R. V. Gr!mkrelidze, On sliding optimal regimes. Dokl. Akad. Nauk SSSR. 143, 1962, 1243-1245 (Russian). English translation in Soviet Math. Doklady 3, 1962, 390-395. 4. L. Markus and E. B. Lee, Optimal control for nonlinear processes, Arch. Rational Mech. Anal. 8, 1961, 36-58. 5. M. Nagurno, Uber die gleichmassige Summierbarkeit und i-hre Anwendung auf ein Variation problem. Japanese Journ. Math. 6, 1929, 173-182. 6. L. S. Pontryagin, Optimal control processes. Uspekhi Mat. Nauk 14, 1(85), 1959, 3-20. English translation in Automation Express 2, no. 1, 1959, 26-30..'55

REFERENCES (Concluded) 7. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mai shchenko, The mathiematical theory of optimal processes, Gosudarst. Moscow 1961. English translations: Interscience 1962; Pergamon Press 1964k 8. E. Roxin,;The existence of optimal controls, Mech. Math. J. 9(1962), 109-1z19 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, 1, 192-254; (b) Un teorema, di calcolo delle variaoioni, Rend. Accad. Lincei, 15, 1932, 417-423. Opere scelte, 3, 84-91; (c) Fondamenti di calcolo delle variazioni, Zanichelli, Bologna, 1921-23, 2 vols. 10. L. Turner, The direct method in the calculus of variations, A Ph.D. -thesis at Purdue University, Lafayette, Indiana, August 1957. 36

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