THE UNIVERSITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Technical Report No. 23 GENERALIZED SOLUTIONS AS LIMITS OF USUAL SOLUTIONS Lamberto Cesari // ORA Project 024160 submitted for: UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOST-69-1662 ARLINGTON, VIRGINIA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR May 1971

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ADDENDUM IV. GENERALIZED SOLUTIONS AS LIMITS OF USUAL SOLUTIONS Lamberto Cesari In the first part of this appendix we show that the generalized solutions as introduced in (1.8-10) can be uniformly approximated by means of usual solutions. In the second part we give an alternate definition of generalized solutions and we show that these solutions coincide with those introduced in (1.8-10). IV.1 APPROXIMATION OF GENERALIZED SOLUTIONS BY MEANS OF USUAL SOLUTIONS As we pointed out in (1.8) it is relevant that generalized solutions can be thought of as limits of usual solutions, and that in the same time, the value of the functional computed on any given generalized solution be the limit of the corresponding values taken by the functional on usual solutions. Since the infimum j of the functional on the class of all generalized solutions is certainly less than or equal the infimum i of the functional on the class of usual solutions, we then are able to conclude that actually j = i as pointed out in (1.8). We shall prove the possibility of approximating generalized solutions (and corresponding values of the functional) for the canonic (Mayer) problem. The analogous result for Lagrange and other problems will then follow as a corollary because of the usual transformations of one type of problems to the other (1.2). We shall use below the usual notations for Mayer problems. Thus, A, 1

U(t,x), M, f(t,x,u) = (fl,...,f ) are given as usual. To simplify matters, we may disregard the usual set B of the t ixt2x -space E +2 or what is the same, assume for a moment that B = E. Actually, we shall keep the first m+~2 point of our trajectories fixed, say (tl,l ) is a fixed point, (tl,x ) E A, and we shall also keep the terminal point t2 fixed, so that we may simply take for B the set B = (t ) x (x ) x (t2) x E. Then the usual function g(t1,xl,t2,x2) shall be considered as a given function of x2 only in E. Then the functional has the form I = g(x(t2)). t Note that for Lagrange problems with functional I = t f dt, reduced to 10 0 Mayer problems by the usual additional variable x, differential equation dx /dt, and initial value x (tl) = O, then we have a new state variable x = 0 0o 1 n (x,x) = (x,x,...,x ), and the functional takes the Mayer form I = x (t ), exactly in the frame of the assumption above. (IV.l.i) Let A be closed, let U(t) be a closed set independent of x and let M = [(t,x,u) (t,x,) c A, u E U(t)] be closed. Let f(t,x,u) be continuous on M and locally Lipschitzian with respect to x in M, and let g be continuous. Let y(t), p(t), v(t), t1 < t < t2, be a generalized admissible system, whose (generalized) control function p is bounded, and whose trajectoryy lies in the interior of A. Then there is a sequence xk(t), Uk(t), tl < t < t2, k = 1,2,..., of (usual) admissible pairs with x (t ) = y(t ), such that xk y as k> k 1 1 k uniformly in [t,t2], and then consequently g(xk(t )) g(y('t2)) as k oo. Proof. It is not restrictive to assume t =, t = b. Let 2d be a 2

positive number which is less than or equal to the distance of the graph G = [t,y(t)), 0 < t < b] from the boundary of A. Let A be the closed d-neighborhood of G. Then G and A are compact, G lies in the interior of A and A c A. 0 0 0 Since v(t) = (u()t), j = 1...,y) is bounded,.there is some N > 0 such that Ju(j)(t)l < N, 0 < t b, j = l,...,7. Let M be the set of all (t,x,u) with (t,x) c A, 0 < t < b, u E U(t), JuJ N. Obviously M C M, and M is a compact subset of E (as the intersection of M closed with the compact 1+n+m set [(t,x,u) (t,x) e A, J|u < N]. By hypothesis f(t,x,u) is continuous on M, and locally Lipschitzian O with respect to x, hence bounded, say If(t,x,u)| N1, and there is a constant L > 0 such that If(t,x,u) - f(t,y,u) < LJx-yl for all (t,x,u), (t,y,u) < M. Given ~ > 0, let 1 > 0 be any number such that ~1 < min [d,e], and take a > 0 so that 3 5 e ~. Let 1 = (2b) a and let 51 > 0 be a number such that |f(t,x,u) - f(t',x',u')j K 1 for all (t,x,u), (t',x',u') E at a distance <. - 1 Let r2 = (2N) ao. Since the y functions u(j)(t) are measurable in [o,b], there is a closed subset K of [o,b] with meas K > b - '12 such that the y functions u(j)(t), j = l,..,y, are continuous on K. Then, K is compact, and the y functions u(j(t), j =,...,y, are uniformly continuous on K. Then there is some 82 > 0 such that lu(J)(t) - u(J)(t')J < for all j = l,...,y and all t, t' E K with It-t' 52. Also, we can take 52 < so small that 2 2- 1 t,t' c [o,b], It-t'l < 2 implies jy(t) - y(t')I < 5 Let us divide I = [o into k equal consecutive intervals Let us divide I = [o,b] into k equal consecutive intervals, say I,...9 ~ks 3

s = l,...,k, each of length b/k. For each Ik, let [ j, J = 1,...y7] be any subdivision of Ik into y measurable disjoint subsets S j c Ik (for instance, subintervals) such that meas Z ksj = f p (t) dt, j = 1,...,. (IV.1.2) ksj Iks j Then Z. meas Z =.I pj(t) dt= I dt = meas Ik ksj j Tks JIks We now take u(t) = uj)(t) for all t c j = l,...,7, s =,...,k. ksj' (Tv.. 13) Then u(t), tl < t < t2, is a measurable function in I with values u(t) E U(t), tl < t < t2, and lu(t) l < N. Let us consider the differential system dx/dt = f(t, x(t), u(t)), 0 < t < b, (IV.1.4) with initial value x(O) = y(O) = x. Since (o,xo)is an interior point of A, the solution x(t) of (III.1.4) exists in a right neighborhood of t = 0, say [O,t], and (t,x(t)) c A Let k be the smallest integer such that b/k2 < 2. Hence, for k > k, and for any two points t, t' E I = I n K, we have It-t' < b/k < 5, and -kso _ I, =v =-' ~ks, knK lu(j)(t) - u()(t') < 1 j = 1,.,v. Let Iko Iks n K, ksjo ksj n K I' = Ik - K, 'ksj = k - K, K' = I - K. For any triple k,s,j with meas ks ks ksj ksj ' S. > 0 we select one point T = T. If meas E = 0 we do ksjo ks ksjjo ksjo

not select any point, and actually we disregard the corresponding term in the lines below. Obviously, meas K' < r2 For any interval [O,t], 0 < t < t, let Kt K' be the sets Kt = K n [O,t], K' = [0,t] - K. Note that JIks Pj( )d - meas Es = 0, and hence Zsj Z f Pj()d - meas Eksjo ks 0 = l( j Iks pj(5)dS - meas ksj) J I' ks pj( )d+ + meas ks j 3 ksj < 0 + S1 f I Pj()d< + s Zj meas Z k < 2 meas K' < 2 2. Ns ks tj s j ksj - - Now x(t) and y(t) are AC with x(O) = y(O) = x, and OY dy/dt = h(t,y(t),p(t),v(t)) =.pj.(t) f(t,y(t),u()(t)), J J 0 < t < b, dx/dt = f(t,x(t),u(t)), O < t < t, x(O) = y(O) = x0. 0 For any t we have u(t) E U(t), hence f(t,y(t),u(t)) is defined, and y(t) - x(t) = Jt [h(,y( ),p( ),v(C)) - f(,x( t),u( )) dt Y( t)- X( ) = t = i t[f(,Y( ),u()) - f(t,x(t),u( )) ]d + ( t + JK' )[ Fjpj( ) f(ty('),u (0)) - f( 'y( t),u(t))]dt = 4 1 + CL2 + 4~3. (Iv.1.6) 5

Since If < N in M we have 31 I [p p(0 f(y(Oiu (0)) - f(,fy([).u())]d) _ JK. [ j pj() If y( )u ()) + If( y(0,u(t)]dl t < 2N meas K' < 2N2 = c Also, we have 11 I = If Z jPj () f(gy( ),u(J)(r)) - f(,y(t),u(t))3dd 2 K j jks = I Z (jt Pj() f(S(,u)())d - f) f(,y()Ou^ i(t)dt( l T Ikso ksj = IZ Ejf(T,y(T),u( (T)) [f p (p)d - meas Zks]l S I kso ksjo Sj I Pj()[f( a y( 5),u t) - f(Ty( T),U (jT)) ]d kso + Z Z J [((O)u( ) ) f TY( j u)())d s ] E ksjo where summations and integrations are extended only over those terms and intervals concerning [O,t]. In each bracket of the second and third sums of the last member we have |-T| < b/k < 62 - < 1' hence Iy() - Y(T)l < 1, u(j)() - (j)(T)( < S1' and each bracket has absolute value < i1. By using this remark and (III.1.5) we obtain hI21 < N ESE lIf pj(d)d- - meas Eksj 2 s IIkso j( ksjo 2 N+ 2 olb =. + = 2. d 2 N2 + 21b = a + = 2a. 6

Finally, l11 < Lf Ljy( ) - x()Ildj. Thus, (III.1.6) yields ly(t) - x(t) _< L t |y() - x( ) ld + 3a, and by Gronwall's lemma, ly(t) - x(t) I < 35 et < 3 eb < 1 < min [d,e], and (t,x(t)) e A for all 0 < t < t. Thus, x(t) is defined in all of [O,b], or t = b, the entire graph [(t,x(t)), 0 < t < b] of x lies in A, and x(t) = xo + t f(5,x(0),u(t))dt, u(t) c U(t), Ix(t) - y(t) I < min[d,e], 0 < t < b. In particular |x(t2) - y(t) | < min [d,~], and (tl,x(tl),t2,x(t2)) e B. Since E > 0 is an arbitrary number, statement (III.l.i) is proved by taking E = k, k = 1,2,.... Statement (IV.l.i) has been generalized so as to include control spaces U(t,x) depending on both t and x, provided f and U vary not too mildly as t and x vary. Namely, let us consider the two following hypotheses: (p) Given N > 0 there is another constant L > 0 such that If(t,x,u) - f(t,y,v) I < L( fx-yl + lu-vj) for all (t,x,u) E M, (t,y,v) e M with -N < t < N, I|x, lyl < N, J|u, Ivr < N. If U depends only on t, or U = U(t), then we require only If(t,x,u) - f(t,y,u) < Lix-y| for all (t,x,u), (t,y,u) c M with -N < t < N, IxI, lYf < N, lu < N. 7

(q) Given N > 0 there is another constant H > 0 such that for any two points (t,x) c A, (t,y) c A, -N < t < N, JIx lyl _ N, and any u C U(t,x) with lul < N, there is at least another point v E U(ty) with lu-vl. HJx-yj. (IV.l.ii) Let A be closed, U(t,x) which may depend on both t and x, and let M = [(t,x,u) (t,x) c A, u E U(t,x)] be closed. Let (f(t,x,u) = (fl,...,f ) be continuous on M, and let g be continuous. Let us assume that conditions (p) and (q) hold. Let y(t), p(t), v(t), t < t < t2, be a generalized admissible system, whose control function is bounded, and whose trajectory y lies in the interior of A. Then there is a sequence xk(t), uk(t), t < t < t k = 1,2,..., of usual admissible pairs with x (tl) = y(tl) and such that Xk + y as k -* co uniformly in [tl,t2]. Remark 1. We have mentioned above that the so called Gronwall's lemma: (IV.l.iii). If u(t) > 0, v(t) > 0, 0 < t < +oo, are given functions, u(t) continuous, v(t) L-integrable in every finite interval, if for some nonnegative constant C we have u(t) < C + ot u(a) v(a)da, t > 0, (IV.1.7) then we have also u(t) < C expt v(a)da, t > 0. (IV.1.8) Proof. If C > O, by algebraic manipulation of (IV.1.7), we have uv/(C + u v d ) < v, 0 8

and, by integration, log ( + f u v d a) - log c < f v(a)da or t t u < C + u v d a < C exp f v(a)dca. 0~~ o~ - ~ 0 If C = O, then (IV.1.7) holds for every constant C1 > 0, and then we have 0 < u(t) < C1 exp f v(a)da, t > O. This relation, as C1 - 0 implies u(t) O. Thereby (IV.l.ii) is proved. Remark 2. In statement (IV.l.i) the hypothesis that the graph G of y be in the interior of A seems to be requested by the proof, since we obviously need a neighborhood of G on which to define the sequence of usual trajectories approaching y. Actually, it is easy to prove by an example that statement (III.l.i) may not be true without the hypothesis that G is in the interior of A. Take n = 3, m = 1, A made up of the three segments joining the points 0 = (0,0), 1 = (2,0), 2 = (1,1) in the xy-plane and then take A = E x A x E1. 0 Let U be the fixed set made up of the two points u = 1 and u = -1, let xyz be the state variables, and take differential system x' = 1, y' = u, z = y, fixed initial point P1: tl = 0, xl = O, Y1 = 0,z = 0, fixed terminal time t2 = 2, B =(P1)x (2) x E3, functional I = z(t2). Then necessarily x(t) = t, 0 < t < 2, and x(t2) = 2. This problem has only one admissible usual strategy u, with u(t) = 1 for 0 < t < 1, u(t) = -1 for 1 < t < 2, the corresponding trajectory in the xy-plane isthe polygonal line 021, and I = z(t2) = 1. The 9

problem has only one admissible generalized (not usual) strategy v, with u()(t) = 1, u( t) = -1, pl(t) = p (t) = 1/2, 0 < t < 2, the xy-trajectory is the segment 01, and I = z(t2) = O. Clearly the only generalized (not usual) solution cannot be approached by means of usual solutions. Here A has no interior points. Remark 3. Unbounded control functions v(t) = (u(),...,u(7), t C t < are allowed in statements (IV.l.i) and (IV.l.ii) under additional dypotheses, which are analogous to those used for a similar purpose in the proof of Pontryagin's necessary conditions with unbounded controls [see vol.l, App. A]. Namely, statements (IV.l.i) and (IV.l.ii) still hold for v(t) unbounded, under the additional assumption (5) There is some number 5 > 0 and an L-integrable function S(t), tl s t < t2, such that t e [tl,t2], Ix' (t, - x" - x(t) | < B, j = l,...,y, implies If(t,x',uJ (t)) - f(txu() (y) Ix' - x"lS(t). As mentioned, the statements above (IV.l.i) and (IV.l.ii) for Mayer problems can be immediately worded for Lagrange problems. For the sake of simplicity we limit ourselves to statement (IV.l.i). Here A, U(t), M are as above, f (t,x,u), f(t,x,u) = (f1,...,f ) are defined on M. Admissible pairs x(t), u(t), tl < t < t2, are defined in (2.9, first paragraph), and the functional is now t I[x,u] = It f (t,x(t),u(t))dt. 1 o 10

(IV.l.iii) Let A be closed, let U(t) be closed sets independent of x, and let M be closed. Let f (t,x,u), f(t,x,u) = (fl...,f ) be continuous on M and locally Lipschitzian with respect to x on M. Let y(t), p(t), v(t), tl < t t, be a generalized admissible system, whose (generalized) control function v is bounded, and whose trajectory y lies in the interior of A. Then there is a sequence of (usual) admissible pairs xk(t), uk(t), tl < t < t2, k = 1,2,..., such that xk - y as k + co uniformly in [tl,t2] and such that I[xk,uk] - I[x,p,v]. 0 By introducing the additional variable x, the differential equation dx /dt = f, the initial condition x (tl) = 0, and new space variable o 1 n = (x,x) = (x,x,...,x ) and function (t,x,u) = (f,f) = (f,f,..,f ), the problem is reduced to a Mayer problem to which (IV.l.i) applies. IV.2 AN ALTERNATE DEFINITION OF GENERALIZED SOLUTIONS (a) Weak Solutions We shall denote by U any fixed control space, that is, any arbitrary fixed set U of elements u, and we shall assume that U has a topology, so that U is a topological space. We shall then denote by (0) the set of all realvalued continuous scalar functions 0(u), u E U, which are continuous on U. We shall denote by M(0), 0 C ([), any real valued functional such that (ml) M is linear, that is, 01 02 c (0), a, P real, implies M(a0 + 2) = aM(1 ) + PM(02); (m2) M is nonnegative, that is 0 E (0), 0 O0, implies M(0) > 0; (m3) M(1) = 1, that is, if 0 denotes the function 0(u) = 1 in U, 11

then M(0) = 0 = 1. The following further properties are consequences of these assumptions: (m4) 01' 02 (0), 0 < 02 implies M(01) < M(0 ); (m5) IM(p) I < M( |I|); (m6) IM() | I Sup 101 where Sup is taken in U, (m7) 0k' 0 Ec ), k = 1,..., k - as k - oo uniformly in U implies M(Ok) - M(f). We shall denote by ~ the class of all real valued functionals M(0) as above. We are now in a position to define the concept of weak solution. As in all available expositions of the theory we assume that U is a fixed set. Let A be any closed subset of the tx-space E1 x En, let U be an arbitrary subset of a Banach space E, and let M = A x U. Let f(t,x,u) = (fl,...,f ) be a vector function defined in M. We shall consider systems (x(t), tI < t < t2, D, M(t,)) where (a) x(t) = i n (x,...,x ) is an AC vector function on [t1,t ]; (c) D is a measurable subset of [tlyt2] with meas D = t2-tl; (d) for every t c D, M(t,0) is a real-valued linear functional defined on (0), that is, on the class of all continuous realvalued function 0(u), u C U, and for every t c D, M(t,0) satisfies the properties (m) above; (e) x'(t) exists and is finite at least everywhere on D (hence, a.e. in [tl,t2]), and dx /dt = M(t;f.(t,x(t),u)), i = l,...,n, t e D, (IV.2.1) or briefly dx/dt = M(t,f(tx(t),u)), t c D. (IV.2.2) Any such system (x(t), D, M(t,0)) is said to be a weak solution, and x(t) a 12

weak trajectory. It is not necessary to indicate the interval of definition of x(t) since [tl,t2] = cl D. It is important to show that every usual admissible pair x(t), u(t), t < t < t2, is a weak solution. Indeed, if we take M(t,o) = O(u(t)), then (IV.2.2) reduces to dxi/dt = fi(t,x(t),u(t)). Analogously, any generalized system (x(t),p(t),v(t)) with p(t) = (P.O,p), v(t) = (u(1),...u( ), v > n+l, is a weak solution. Indeed, if we take M(t,o) =Z.p.(t)(u(j)(t)), then (IV.2.2) reduces to dxi/dt = jp (t)fi(t,x(t),u ()t)). Thus a weak solution appears to be an extension of both usual and generalized solution. Actually, under mild hypotheses, every weak solution is a generalized solution, and is the weak limit of a sequence of usual solutions, (b) Weak Solutions as Quasi-solutions and Generalized Solutions We shall denote as in Chapter 1 by R and S the sets R(t,x) = co Q(t,x), S(t,x) = cl co Q(t,x). Equations (IV.2.1), or (IV.2.2), can be thought of as defining a director field, namely dx/dt e Q*(t,x), where Q* is the subset of E defined by n Q*(t,x) = (M(t,f(t,x,u)) |M e ) c En, that is, the subset of all vectors M(t,f(t,x,u)) = (M(t,fl), M(t,f2),...,M(t,fn)},

when M describes the family > of all possible real-valued linear functionals M satisfying properties (m). (IV.2.i) For any space U, we have co Q(t,x) c Q*(t,x), that is R(t,x) c Q*(t,x). Proofs For every u c U, let M(^) = O(u), in other words, let M be the Dirac operator 5- at u, which gives for every c ({) the values of 0 at u. U Then M(f(t,x,u)) = f(t,x,u) C Q(t,x), and as u describes U, then f(t,x,u) describes Q(t,x). Thus Q c Q*. Now 3 has a linear structure, namely, if Ml, M2 e ' and 0 < a < 1, then M = a My + (1-a)M2 E. Thus M(O) = Ca (0) + (l-a)M2(0), and we conclude that Q* is convex, and hence Q* contains the convex hull of Q. (IV.2.ii) If U is any metric space, then co Q(t,x) c Q*(t,x) c cl co Q(t,x) Proof. For each point u E U let G = G(u) be a neighborhood of u where f(t,x,u) as a function of u above has an oscillation < e, say If(u') - f(u")[ < ~, for all u', u' c G = G(u). Then the collection (G) of all these neighborhoods is a covering of U, that is, the union of all sets G is U, say UG = U. Then we know (see Remark below for references) that U possesses a partition of unity, that is, U possesses the following important property: Given (G), there is another covering (G', i E I), I an index set, and certain scalar functions a.(u), u E U, i E I, with a.(u) ~ (0), such that (a) UG. = U as 1 3-i~~ 1 14

above; (b) each open set G. is completely contained in at least one set G c (G); (c) (Gi, i c I] is locally finite, that is, for every u e U there is a neighborhood V of u in U such that V n G / 0 for at most finitely many i; (e) 0 < ai(u) < 1 for all u c U and i c I; (f) Z ai (u) = 1 for all u E U. Note that for each given u E U the sum in (f) is actually a finite sum: indeed, if V is the relative neighborhood as in (c), then V n G. / for at most finitely many i, and thus ai(u) = 0 for all remaining i e I, and the sum.iI ai(u) possesses at most finitely many terms different from zero (at any given u). Let u. denote any point u. E G., i c I, and let g denote the function g(t,x,u) = iI f(t,x,ui) a1(u). Then, if u E U and ai(u) L 0 for some i e I, then u c G., u. e G. c G for some G e (G), and f(t,x,ui) = f(t,x,u) + G., le. <. Also Ig(t,x,u) - f(t,x,u)l = IZif(t,x,u.) Oi(u) - f(t,x,u) -= iZ.i a.(u)I < E..(u) = ~, or Ig-f| < s for all u E U. Let Pi = M(a.(u)), i e I. Then, by properties (m) we have 0 < p. < 1. Also we have 1 = M(1) = M( ii(u)) = ZiM(ci(u)) = P. On the other hand JM(g(t,x,u)) - M(f(t,x,u))| < ~, M(g(t,x,u)) = M( Zif(t,x,u)ci(u)) = Zif(t,x,ui)M(ai(u)) ZiPif(t,x,u i) co Q(t,x). 15

Thus, every point M(f(t,x,u)), that is, every point of Q*(t,x), is a point of accumulation of points of co Q(t,x). Thus co Q(t,x) c Q*(t,x) c cl co Q(t,x). (IV.2.iii) If U is any metric space, then whenever R(t,x) is a closed subset of E, then Q*(t,x) = R(t,x) = S(t,x). In particular, this is certainly the n case if U is compact. Indeed R closed implies R = S and hence Q* = R = S because of (ii). If U is compact, then certainly Q = f(t,x,U) is compact, and then R is compact and hence closed, and the statement applies. We conclude now with the following theorem: (IV.2.iv) If U is any metric space, and (x(t),D,M) is a weak solution, then x(t), t < t < t2, is a quasi-trajectory, that is, an AC solution of the orientor field dx/dt E S(t,x(t)), S(t,x) = cl co Q(t,x) = cl R(t,x). If, in addition, R(t,x(t)) is known to be closed for almost all t e [tl,t2], in particular, if U is compact since then R = S is also compact, then x(t), t < t < t2, is a generalized trajectory, that is, there is a probability distribution p(t) = (P1,.l.,P ), v > n+l, and a vector function v(t)= (u (),...,u ), with p.(t), u()(t) measurable in [tl,t ], u()(t) c U a.e. in [ti,t2], 0 < p (t) < 1, j p.j(t) = 1 for all t c [ti,t2], dx/dt = j=1 Pj(t) f(t(t),u (t)) a.e. in [tl,t2]. 16

Proof. The first part is obvious since dx/dt e Q*(t,x(t)) c S(t,x(t)) because of (ii). If R (t,x(t)) is closed for almost all t, then R = S, and hence dx/dt C R(t,x(t)) for almost all t c [tl,t ], where R(t,x) = co Q(t,x) c co f(t,x,U), and here U is a fixed set which, therefore certainly satisfies the conditions of Theorem (1.6.i) of Chapter 1. If h(t;x,p) = Z pjf(t,x,u(i)), p c P, v E U, as in Chapter V, then we have also R(t,x) = h(t,x,r x U ) and again r x U is a fixed set. By the Theorem above, we conclude the proof of our statement. Remark. Both statements (ii) and (iii) above certainly hold for all topological spaces U for which a partition of unity property holds, thus, in particular, for all paracompact spaces U, more particularly, for all metric spaces U [N. Bourbaki, Topologie Generale, Ch. 9, Sec. 4, no. 3,4,5]. A collection (G} of subsets of a topological space U is said to be an open covering of U if the union of all G c (G) is U, say UG = U. Another collection (G') of open subsets of U is said to be a subcovering of (G) if UG' = U, and if for every G' e (G') there is at least one G e (G) with G' c G. A covering (G) of U is said to be locally finite provided, given u E U, there is some neighborhood V of u in U such that V n G / 0 for at most finitely many G E (G). A topological space U is said to be Hausdorff if for any two distinct points ulu2 e U there are open sets G1, G2 in U with u1 c G1, u2 c G2, G1 G2 = 0. Also, U is called normal, if for any two closed subsets H1, H of U with 17

H n H2 = 0, there are open sets G1, G2 in U, with HI c G1, H2 c G2, G1 n G = 0. Finally, a topological space U is said to be paracompact provided U is Hausdorff and every open covering of U possesses a locally finite subcovering. Then, every metric space is paracompact [N. Bourbaki, Ibid., Ch. 9, Sec. 4, no. 5, Th. 4]. Every paracompact space is normal [Ibid., Ch. 9, Sec. 4, no. 4]. Every paracompact space possesses the partition of unity property [Ibid., Ch. 9, Sec. 4, no. 4, Cor. To Prop. 4]. (c) Limit of Weak Solutions Any two weak solutions, or systems (x(t), D, M}, (x(t), D', M') are said to be equivalent (or identical) provided they have the same trajectory x(t), tl < t < t2, [tlt2] = cl D = cl D', and provided, given any vector function g(t,x,u) = (gl,...,g ) continuous on A x U, we have 1 tl x(t) = x(tl) + tl M(T,g(T),x(T),u)dT = 4 M (Tg ),X( )UdT and (t,z(t)) E A for all tl < t < t2. In other words, two systems (x(t),D,M), (x(t),D',M') are equivalent, or identical, provided the vectors obtainable by integration on every function g along the common trajectory x(t) are the same. This definition establishes an equivalent relation, and we shall denote by C* any equivalent class of such systems (x(t),D,M). For the sake of simplicity, we still call C* a weak solution, and any system (x(t),D,M) of the equivalent class, a representation of C*. We shall now introduce in the family (C*) of weak solutions C* a concept of limit. We shall define this concept by using arbitrary representations

(x(t),D,M) for every C*. As we shall see, it is immaterial which of the representations are used, and thus, for the sake of simplicity, we shall word the concept of limit as follows. We say that a sequence [Ck] of weak solutions, say k = [xk(t), tk < t < tk Dk' M(t,], k = 1,2,..., converges to a weak solution C = [x(t), t < t < t2 D, M(t,2)] provided (a) xk converges in the metric p toward x, or p(xk, x) - 0 as k + oo, and hence t lk t, t2k - t2, k(tlk) + x(tl), x(t ) - x(t), as k + o. (b) For every vector function g(t,x,u) = (gl,..,g n continuous on A x U 1 n as f, and initial values, say yk(tlk) = (y k',..Y lk = y 11 such that yk(tlk) - x(tl) as k - oo, and such that, if y (t = 1 + tk M (t,g(t,x(t),u))dt, t < t < t2k k = 1,2,..., k tkl + kk k 1k - 2k t y(t) = x(tl) + f M(tg(t,x(t),u))dt, t < t < t2, with (t,yk(t)) e A for t E [tlk,t2k], (t,y(t)) c A for t c [tl,t2], we also have p(yk,y) + 0 as k - oo. With this definition, the family (C*) of weak solutions is an L-space 19

(see Remark below). For A - E1 x E it has been proved that (C*) is metrizable n (E. J. McShane [68 m3). (d) A Compactness Theorem for Weak Solutions We shall now prove that, under conditions on A and U, every sequence of usual solutions possesses a subsequence which "converges" toward a weak solution. This statement is included in the following theorem, where we show that every sequence of weak solutions possesses a subsequence which converges in the sense of the previous section, toward a weak solution. For generalized solutions a compactness theorem was proved straightforwardly in.(2.7). (IV.2.v) If A is a compact subset of the tx-space E1 x E, if U is a fixed n compact topological space, and f(t,x,u) = (fl,...,f ) is a continuous vector function on M = A x U, then the set of all weak solutions is compact in the L-topology. We shall precisely prove that, given any sequence of weak solutions (xk(t)j tlk< t < t2k, Dk, M(t,~)), k = 1,2,..., there is always a subsequence which converges in the L-topology toward a weak solution (x(t), t < t < t, D, M(t,O)) and correspondingly the weak trajectories xk converge in the p-metric toward the weak trajectory x. Proof. The set M is compact and hence there exists a constant M such that (t,x,u) E M implies -M < t < M, Ixl < M, lu < M, and also 0- -0 - 0 - 0 |f(t,x,u)I < M. Then, property (m6) implies M(t;f(t,x(t),u)) < M, hence, -. o for every weak trajectory x(t), t < t < t2, we have!dx/dtj < M a.e. in [tlt2], and x(t) is Lipschitzian with constant M. Also (t,x(t)) e A for 0 20

all t e [t,t2 ] where A is compact. Thus, the weak trajectories x(t) are equibounded, equicontinuous, equilipschitzian. Given the sequence above (xk(t), tlk < t < t2k, Dk, M(t,0)), then by Ascoli's theorem there is a subsequence of weak trajectories, say still [Xk], which converges in the metric p toward a continuous vector function x (t), tl t < t2, and x (t) is Lipschitzian with constant M, hence AC in [tl,t2]. Since A is closed, (tk, xk(t)) E A implies (t,x(t)) E A for all t E [tl,t2]. We have lk(t,f(t,xk(t),u))l < M for all t c Dk, with Dk c [tlkt2k] meas Dk = t2k-tlk. If CDk denotes the complement of Dk in [tlk,t2k] and we take D' = (t,t2) - Uk CDk, then D' is a measurable subset of (tl,t2) with meas D' = t - tl. If Jk(t) tk < t < tk, denotes the function k(t):= ft. (t,f(t,x (t),u))dt, tlk t t2k k 12,... *k~t) l k t k- k - then 0k(tlk) = 0, and for all t,t' E [tlk,t2k] also |lk(t) -!k(t') < M It-t'l, JIk(t) < 2M2. Thus, *k(t) is a sequence of equibounded, equicontinuous, equilipschitzian k functions. By Ascoli's theorem there is a further subsequence, which we still denote [|k], and which is convergent in the metric p toward a continuous function r (t), tl < t < t2, and * (t) is Lipschitzian with constant M and 0 0 2 — 0 hence AC in [tl,t2]. We shall now consider the class (F) of all scalar functions F(t,x,u) which are continuous on M = A x U. Obviously, there is a sequence F, S 21

s = 1,2,..., of such functions such that, given any F of the same class and ~, 0 < s < 1, there is also some F in the sequence with IF(t,x,u) - F (t,x,u ~ for all (t,x,u) e M. For every s we shall consider the scalar functions 4ks(t) = It Mk(t,F(t,xk(t),u))dt, tlk t t, k = 1,2,.... Since IF I < N on M for some constant N, then for all t,t' E [tlk,t2k] we have ks(t =, lks(t) - k(t')l < N Itt'', (t)l 2M N ks 1k ks ks ks 0 s Thus, by Ascoli's theorem and the usual diagonal process, we can perform the selection of a subsequence, so that 4k converges as k -Y oo in the metric p ks toward a continuous function ks (t), tI < t < t2, which is then Lipschitzian with constant N in [t',t2] with fos(tl) = 0, and this holds for every s 1 2 Os 1 s = 1,2,.... For every s, the AC function o (t) has finite derivative 4' (t) at all Os Os points t of a measurable set D' c [t,t2] with meas D' = t-tl, and we take s 12 s 2 1 as before D" = (t,t ) - U C D', D = D' n D", and then D (tIt2), meas D 12 s s 2 t t 2-t1 We have now chosen a well determined subsequence, which we still denote by [k], and t t, t2 t2 as k -> (along the chosen sequence). Thus t and t are also well determined. Given F e (F) let us take k(t) = Itl Mk(t,F(t,xk(t),u))dt, tlk < t< t k = 1,2,..., k 1 -- t klk - -_2 22

and let us prove that *Vk converges in the metric p toward some r(t), tl < t < t2, which is AC in [tlt2]. Indeed, IF(t,x,u) I < N for some constant N and hence ~k(tlk) = 0~ I[k(t) - 'k(t')l I Nt-t' I It(t) I 2M N, *k (t lk O''k k -k 0 (IV.2.1 ) for all tt' c [tlk t2k], k = 1,2,.... On the other hand, given e > 0, there is some s such that IF(t,x,u) - F (t,x,u) [ < e for all (t,x,u) c M, and also there is some k such that k,k' > k implies p(rk,V' ) < ~. Now o -o ks ks lk(t) - ks(t)l = It lMk(tF(t,kt),u))dt - t (tF (tx),) )dt tl IMk( (tF(t, t),u)) - Mk(t,F (t,x(t)u)) dt = tl IMk[tF(tx(t),u) - F(t,xk(t),u)]ldt < 2M, and an analogous relation holds with k replaced by k'. Thus kP( *kk) < P(kk' ks) + P(ks,'k, s) + P(k's, k' < (2M +l), for all k,k' > k. This proves that k (t), k = 1,2,..., converges in the -o k metric p toward a continuous function r(t), t < t < t2. Also (IV.2.1 ) implies, for every t,t' e [tl,t2], (t ) = 0, I|(t) - r(t')J < NJt-t'l, I|i(t) < 2M N. 1 o~~~~~~~~~~~~~ 23

Now let us prove that, for any F E (F), the corresponding function r(t), t < t < t, so determined, has finite derivative v'(t) at every point t E D, 2 21 where D is the subset of (tl,t2) determined above with meas D = t2-t1. Indeed, given F e (F), let us choose F as above and note that, s F(t,x,u) - F (t,x,u) - ~ < 0, F(t,x,u) - F (t,x,u) + ~ > 0 for all (t,x,u) c M. Hence, Mk(t,F(t,xk(t),u)) - M(t,Fs(t,xk(t),u)) - 6 < 0 < Mk(t,F(t,xk(t),u)) - M(t,Fs(t,xk(t),u)) + ~ a.e. in [tlkt2k], and by integration we conclude that the scalar functions tk(t) - ks(t) - ~( t-tlk), ks(t) + ~(t-tlk) are monotone nonincreasing and monotone nondecreasing respectively. Passing to the limit as k + oo, we obtain two scalar functions (t) - (t) - ~(t-tl), (t) - (t) - ost) + (t-t) which have the same property. If we denote by D, D the usual lower derivative and upper derivative operators, of a scalar function, we obtain D(t- ' - <, D(t) - J' (t) + E > 0 aevOSr- Os - at every point t c D, since D c D and 4 has derivative in D. Thus, s os s 24

0 < D5(t) - Dv(t) < 2E, t D c (tl't2), where D is a fixed set and e is arbitrary. We conclude that t' = Dh = Dr exists and is finite at every t c D. In other words, for every F E (F), the AC function defined above, say r(t) = r(t,F), tl < t < t2, has finite derivative r'(t) at least at every point t c D c (tlt2) with meas D = t2-tl. We shall define M(t,0) by taking M(t,F(t,x(t),u)) = (d/dt)v(t,F), t c D, F c (F). The properties (ml,2,3) for M can be verified immediately. In addition, since f(t,x,u) = (fl,...,f) and each f. is a scalar function of the class (F), we l '1 n i have dx /dt = (d/dt)t(t;f.) = M(t,f(t,x (t),u)), t E D, i = l,...,n. We conclude that (x (t), tl < t < t2, D, M(t,0)) is a weak solution. Finally, assume that g(t,x,u) = (gl,...,g ) is any continuous vector 1 n function in A x U, that yk(t) = (yk...yk) tlk < t < t2k k = 1,2,..., are AC vector functions satisfying (a) (t,y (t)) E A for all t c [tlk,t2k], (b) dyk/dt = Mk(t;g(t,xk(t),u)) a.e. in [tlkt2k], 1 n 1 n and (c) Yk(tlk) = Ylk = (yl (Y,.y) = E as k- 'o. Then tlk t t2k + t2, and for each i = l,...,n, the function gi(t,x,u) is a scalar function of the class (F), and the differences 25

t < I < t k k(t ) - yk(t) - Ylk' 1k 2k are functions *k above, and hence converge in the p-metric toward an (AC) function v(t), tl < t < t2. If y(t) = (y,...,y ), y(t) = t (t) + y, i = l,...,n, then p(yk,y) + O, as k + o, dy/dt = M(t,g(t,x (t),u)) t c D (t,t), J(xkDk, M;glk) = (tlkYk(tlk),t 2kYk( 2k) (t,y(tl)t2,y(t2)) = J(x oD,M;g,y) Thus, (Xk,Dk,M) converges in the L-topology toward (x,D,M). Theorem (IV.2.v) is thereby proved. Bibliographical notes. Statement.(IV.l.i) was observed by R.V. Gamkrelidze [48 a]. The present proof is by Cesari [24 m], who proved also the analogous statement (IV.l.ii) for the case in which U, the control space, depends on both t and x [24 m]. The material of Section (IV.2) is in L. C. Young's [121] and E. J. McShane's papers [77 m,n,o,p]. For a recent discussion of generalized solutions and the question of approximating them by means of usual solutions, see J. Warga [114]. 26

UNIVERSITY OF MICHIGAN III1I l nBll /111 I1III I IIIIIIIII III //11 3 9015 02844 9380 ENGIN.. TRANS. LIBRARY 312 UNDERGRADUATE LIBRARY -764-7494 OVERDUE FINE., 25~ PER DAY DATE DUE '' ~ ~~~~ - zw~ ~f~,il —.~r~- 14 ~ - CC I 0*C '