THE UNIVERSITY OF M I C H IGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report No. 20 UPPER SEMTCONITINIJITY PROPERTIES OF VARIABLE SErS IN OPTIMAL CONTROL David E. Cowles ORA Project 02416 submitted for: UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOSR-69-1662 ARLINGTON, VIRGINIA administered through: OFFICE OF RESEARCH ADMINISTRAT'ION ANN ARBIOR February 1971

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UPPER SEMICONTINUITY PROPERTIES OF VARIABLE SETS IN OPTIMAL CONTROL* David E. Cowles 1. INTRODUCTION In his existence theorems for optimal solutions in control theory, Filippov [1] used the concept of metric upper semicontinuity of subsets of Euclidean r +1 spaces E as basic requirements for compact equibounded control spaces. Cesari [2, 3, 4], Lasota and Olech [5], Olech [6,7], and other authors have shown that, when the control space is only closed and not necessarily bounded, then either Kuratovski's concept of upper semicontinuity (property (U)), or Cesari's variant of this concept (property (Q)), are more suitable. In the present paper (E 2) we introduce a scale of intermediate concepts, or property Q(o) for p any integer, 0 < p < r + 1. We prove then (E 2) that property Q(p) for p = 0 reduces to Kuratovski's property (U), and for p = r + 1 reduces to Cesari's property (Q). In addition, we prove that property Q(p + 1) implies property Q(P), 0 < p < r. In 5 3 we prove further statements concerning property Q(p). We shall show in subsequent papers [8, 9] that the use of these properties Q(p), 0 < p < r + 1, will allow a considerable reduction of the hypotheses needed in lower closure and existence theorems for optimal solutions in optimization problems with distributed and boundary controls. *Work done in the frame of US-AFOSR Research Project 69-1662. This is part of the author's Ph.D. thesis at The UJniversity of Michigan, 1970. 1

2. PROPERTIES (U), (Q), and Q(p) OF SET VALUED FUNCTIONS Let C be a measurable subset of t = (t1,,t ) space Ev, v > 1, and for every t c C, let A(t) be a non-empty subset of y = (y,,y ) space E, s > 1. Let A be the set of all (t, y) E EV x E such that t E C and y E A(t). For r +l every point (t, y) c A, let Q(t,y) be a nonempty subset of the z-space E, z = (z, z.. z )r r > O. For any point (t, y ) c A, and E > 0 we shall O O denote by N (t, y ) the set of all (t, y) e A at a distance < E from (t, yo). Also, given any subset F of a Euclidean space E, we denote by cl F, co F, cl co F the closure of F. the convex of F. and the closure of the convex hull of F, respectively. We state now the definitions of Kuratovski's property (U) and of Cesari's property (Q). These properties have been studied in Cesari's papers [2, 3, 4]. We say that the sets Q(t, y) have property (Q) at a point (t, y ) C A provided Q(t,y) n cl co Q(t,, Y c), o o E>O 0 0 where Q(t, y, U) o o (t,y) E N (to,y)Q We say that the sets Q(t, y) have property (U) at a point (to, yo) E A provided Q(t, yo) = cl Q(t y, E). We say that the sets Q(t, y) have property (U), or (Q), on A provided they have the same property at every point (to, yo) c A. Sets having property (U) are closed, and sets having property (Q) are closed and convex. We now give the definition of property Q(p), 0 < p < r+l. Property Q(p)

is so designed that for p - r+l it is equivalent to property (Q,) and for p = O it is equivalent to property (U) as we shall prove below. For any integer p, 0 < p < r+l, we say that the sets Q(t,y) have property Q(jp) at (t, y ) c A provided for every z = (z,...,z ) ~E r o0o 0 o 0 r+l i i Q(t yQ) N z E t z = z i = p,,r] = 3A cl co ~r~~~ ~E+l il [Q(t, y, c) n {x c Iz - z |<D, i =, *, r]] with E. Also note that, if we denote by P and Q the sets in the first and second members of the equality above then certainly P c Q. Thus, for property Q(p) at the point t, y we actually require that P D Q (for every z c Er+l) 0 0 0' We shall use the following notations. For every z c E, p a nonnegative integer, 0 < p < r+l, and D > O, let r+l Ng(z; p) be the (cylindrical) set of points in E whose final (r+l) - p coordinates are within 5 of those of z; i.e., 0 N (z; P) z E zl <, i- z | <, i Also we denote by N(z; p) the set r+_ i i IN(zO, ) - (Z c Er 2 i,r] r +1 Thus, N (z; O) - N(Zo) N(z; O) = [z ], and N (z, r+l) = N(z, r+l) E r+l For any subset F or E and number > 0O we denote by (F) the closure of the set of points which are within q of a point of F. We refer to (F) as the r+l closed q-neighborhood of the set F in E 3

r+l (2.i) For every c > 0 and P > 0 let Q(c, P) be a subset of E. Then f n Q(,E) = n n Q(c,'). P>O c>O 0>O P>0 Proof: Let z c n n Q(c, e ) Then z E >n Q(c, ) for every fixed C>O 10'' E > 0 and z c Q(e, P) for every P > 0 and each fixed E > O. Now if a property holds for every P > 0 for each fixed e > 0, then the same property holds for every C > 0 for each fixed P > 0. Thus z C Q(E, A) for every ~ > 0 and each fixed P > O. Hence z E A Q(, P) for each fixed P > 0 and z C n0 A Q(c, 1). We have proven that n Q(c, 1) c n n Q(E, 1). The E>0 C>0 1>0 13>0 E>O proof of the reverse inequality is similar. (2.ii) For every (t, y) c A let Q(t, y) be as in the first paragraph of this section. Let p be any integer, 0 < p < r+l. Then for every (t, Y) v+s E. nO no cl co (Q(to, Yo, ) n N(Zo; p)) -= O O Ncl co (Q(t, y, C) N (z; p)), >0:>0 ( y, Proof: This statement follows immediately from (2.i). We now study the relationships between property Q(p) and properties (Q) and (U). (2.iii) Theorem. For every (t, y) E A let Q(t, y) be as in the first paragraph of this section. Then the sets Q(t, y) have property Q(r+l) at a point (t, y ) E A if and only if they have property (Q) at the same point. The sets Q(t, y) have property Q(o) at (t, y ) c A if and only if they have property (U) at the same point.

Proof: The first statement is apparent from the definitions of properties (Q) and Q(r+l). To prove the second statement, let the sets H and I be defined as follows: rr+l H = fn O cl co [Q(t, y, c) n {z C E+ Iz - z < I ]] sI cO cl LQ(to, yo, c) 1 {z e E I z - z_ < }] We need only show that for e-very z c Er o rr+1 Q(t, y ) n (z E z c z } H if and only if Q(t,y ) n (z E E z=z ) I, The last statement holds if H = I. Clearly, I c H. We show that H c I. Suppose that z E H. Then, for every c > 0 and every P > O, there exists a point p c Q(t, y0, c) such that IP - z I <. Hence, for every c > 0 and every P > 0 z C cl[Q(t, y Y e) n (z e E | z - zI < Z). This statement implies that z c I and that H c I. o (2.iv) Theorem. Let Q(t, y) e E be as in the first paragraph of this section, and let p be any integer, 0 < p < r. Then property Q(p + 1) implies Q(p). Proof. We need only show that Q(t, y ) n (Z Eri z = z i = p + i,...,r) 5

I ci co ((t L, y c) f I(z E - p - C>>O ~o o o -- implies that r+l i i...r Q(to, yo) n [( E i z z, P =,r) z c n cl co (Q(to, y,, ) n tZ. E*rs-z <, p,,:. Using property Q(p + 1), we have r+l i i r+1 p P Q(t, y ) n ( i = =, P,,r] = (z E IZ = ) n [Q(to, yo) n (z = z =o, = p + 1,,r}] D {z z E = z ] n ln cl co [Q(to, Y, ) o c>O o o> n (z j Zi z <, i P +,,r]] no (zl IZP- Zl <':} n n n cl co [Q(to, Yo, ) = p 1,r] z z - z < ) 0 n > (c co EQ(to, yo c) n (z z' - i i = p. P>O c>O o o - i — P lj,,r)] n [I %! "P- z ) <P) B~O ~>O ci o D~(t r Y,, Fl n nZj I~ - zco < t, - p,...,nI< This completes the proof of Theorem (2.iv). For every (+, y) c A let Q(t, y) be as in the first paragraph of this section. We say that for a given (t, y ) ~ A and O < p < r+l, the set Q( to, y ) is p-convex provided 0

o0 -1 P r (z0 p- z,..Zz) E Q(t y), 1 2" 2 o1 1o0 0 0 0 implies C z1 + (- () 2 c Q(to, yo) for all 0 < a < 1. In this definition we understand that (r+l)-convexity is r+l the usual convexity of Q(t, yo) in E. In other words, Q(t, yo) is p-convex 0 0 o r r+l provided, for every z =(zo,.,z)) c ), the sets o 0 YOOV r+l i i Q(t y o) n c B I z z..,r are convex. By known properties of convex sets, the set Q(t, yo) is p-convex, if and only if for every t > 1, real numbers \ > 0, \+...+ 1, and points o p-i p p z (z...,z, z,...,z) Q(t, yo), y 1,..., I 7'Y Y 0 0 0 0 we also have ~ % z c Q(t, y 7 7 Y O O (2.v) If the sets Q(t,y) have property Q(p) at (t, yo) c A for some p, O < p < r+l, then Q(t, y ) is p-convex. Indeed, by the definition of property Q(p), the sets r+l i i Q( t y ) n z E z, i = p,., r are closed and convex as intersection of sets which are closed and convex. We now show that property Q(p + 1) is preserved in the sense given below, under addition of a continuous function.

(2.vi) Theorem. For every t E C, y E A(t), let Q(t, y) C E be as in the first paragraph of this section, let p be any integer, 0 < p < r, and *(t), t c C, be a real valued continuous function on C. For every t E C, y c A(t), let Q+(t, y) denote the set Q;(t, y) (z Er+l z = p + (V(t), 0,...,0) for p c Q(t, y)). If the sets Q(t, y) have property Q(p + 1) on A, then the sets Q (t, y) also have property Q(p + 1) on A. Proof. We designate by Q (to, yo, c) the set +- + Q (t, y, c) U (t Y) (t, y) c A n N (t, y ) C 0 We need only show that for every (t, y ) E A and z E,r+l Q(t y) n N(z; p+1) contains the set n n cl co [Qrto, yo, ) n N (zo; + 1)]. Let us take an arbitrary point p in the latter set. Let ~ > 0 be an arbitrary positive number and take c - E ( ) positive so small that 0 0 sup It(t) - (to ) < V/2. (t, y) c A n N (t, y) E 0 0 0 Now for every pair of positive numbers (e, P) with 0 < c < E, there exists a point p, + r+l p C co [Q (t, Y, C) n N(z; p - 1)] C E such that Jp - pI | </2. Then p is a convex combination of r+2 points of

the set is brackets, or r +1 P c Q7t yiPi + fi((t )i) a1 > ir),9 i=o Pi (to, yo, ) n N (zo; p + 1) and 1ki 1, k > 0, i = O,l,...,r+l. Since IP - PO I < r/2, we have Po - (4to),o,...,o) C [C1 co (q(to, y, E) nN(zo; p+l))] (2..13) for every > > O.30, and ~, 0 < c < E. Since Q(to, yo, E) is a subset of Q(t, y, E') for every 0 < e < e', equation (2.1.3) holds for arbitrary c > 0, 5 > 0, and q > 0. Since r is arbitrary and the set inside the brackets is closed, P - ( (to),0,...,0 ) C cl co [Q(to, yo, ) n N(zo; p+l)] for every c > 0 and 5 > 0. Hence P - (t(t ),o0,..,0) E En cl co [Q(t, y, y ) n N(Zo; p+l)] Since Q(t, y) has property Q(p+l) on C, P - (4r(t ),o,...,) E Q(to, yo) n N(z; P+l). Finally this statement implies PO c Q(to, y o) n N(zo; 0+l) and completes the proof of (2.vi).

Te now show that property Q(p) is preserved, in the sense given below, upon multiplication by a positive, bounded continuous function. r+l (2.vi) For every t c C, y c A(t), let Q(t, y) c E be as in the first paragraph of this section, let p be any integer, 0 < p < r+l, and J(t), t C C, -1 be a real valued continuous function with 0 < K < J(t) < K for all t c C and some constant K. For every t C C, y c A(t) let QJ(t, y) denote the set Q (t, y) z C Er+l I z = p J(t) for p e Q(t, y)). If the sets Q(t, y) have the property Q(p) on A, then the sets QJ(t, y) also have property Q(p) on A. r +1 Proof. We only need to show that for every (t, y ) E A and z C E fno cl co (Q (t, y, C) N ( )) (t, y) 5>O c>O J 0 0 o 0 - J o 0 nlN(z; P). Take a point p in the set on the left hand side. Let q > 0 and 6 > 0 be arbitrary numbers. Since i( z i(J(t)) - (J(t )) < (z - z )(J(t )) + iz (J(t)- J(t)) (J(t)J(t))- J, 0 0 for Iz - z i < P and t sufficiently close to t, we may make the difference O O on the left hand side less than or equal to 2KP. Consequently, we can determine a number c = c(~q, 5) > O small enough so that 10

(i) 2( IpoI + 1)K2 sup IJ(t) - J(t )I< q/2, 0 0 (t,y) c A fnN (t, yO) 0 (ii) if z e N (z; p) then z(J(t))- E N1 (z (J(t)) ) for all t c C n (tl it - t I< E~. Given c', 0 < cE' < c0, there exists a point p such that p - mPo _< /2K, plP _< 2( Ipo| +1), and r+l c E co [Qj(t, yo, c') n N (zo; P )] E Then p is the convex combination of r+2 points wj of the set in brackets, or,r+i +1 p = )I._ hjwj; Sf. k.=l, 0. > 0, J-0o J J a- a wj Qj(t, o c') N (z; p ), j = 0,1,...,r+l. That is, wj = qjJ(tj) with qj c Q(tj, yj) and (tj, yj) c NC (to, yO) n A, j = 0,1,...,r,r+l. By choice of c' and the statement (ii) q. 00( tog YO') n (Zo(J(t)); p). Consider the point P r+l r+l P = ( ) )jJ(tj))- ( Z \jJ(tj) qj) j = a0 j=0a a a 11

in the set co[Q(t, y, E) n N K(z(J(t)) ) r +1 Since P = ( J(t ))-1 IP p(J(t ))-ll < J(t)- ) - J(t )I j=j (IJ(t ) ~ jJ(t ) I)-jPI < 2( jIp + 1)K2 sup IJ(t) - J(t ) < /2. (t, y) E A n Nq (to, y) Consequently, IP-po(Jt < - P(j(t- p(J(t ) - P+ I p(Jt ) )) - _< /2 + r/2 =. We have p(J(t ))- Ce [ci co (Q(to, y, c) n N2((Jt 1; )] where u > 0 and P > 0 are arbitrary and c' > 0 is arbitrary so long as CI < C (rl, 3). Since Q(t, y, cE") contains Q(t, y, c') for c" > c', the above statement holds for arbitrary positive r, P, and c'. Also, because of the fact that the set inside the brackets is closed and ~ > 0 is an arbitrary positive number, p (J(t )) C cl co [Q(to, y, E') NN2y(z ( J(t)); for arbitrary c' > 0 and 5 > O. Since Q(t, y) has property Q(.p ) on C, 12

and O E Q(tO, Yo) n N(z; P) 3. A SUFFICIENT CONDITION FOR PROPERTY Q(p) We consider now a situation which occurs often in optimal control theory. Let Abe aclosed subset of the (t,y)-space EV x E, and for every (t, y) c A let U(t, y) be a given subset of the u-space E, u = (u,,l1 ). Let M denote the set of all (t, y, u) with (t, y) c A, u E U(t, y). Let f(t, y, u) = (f,..., f ) be a given continuous vector function on M, and for every (t, y) c A let Q(t, y) be the set Q(t, y) - f(f, y, U(t, y)) r+l r+l [z E I z = f(t, y, u)u c U(t, y)] E. (3.i) Let us assume that A is closed, f continuous on M, the set Q(t, y ) is p-convex for some 0 < p < r+l, and that the sets U(t, y) are compact, uniformly bounded for (t, y) in a neighborhood N6(t, y ) of (to, y ) in A, and have property (U) at (t, y ). Then the sets Q(t, y) have property Q(p) at (to, y ). Proof. Let B denote a cube of the u-space with U(t, y) c B for all (t, y) F N(t, ).y Let M denote the set of all (t, y, u) with (t, y) c N(t, y ), - r+l u e U(t, y). Since M c N(to, yo) x B, M is bounded. Let z c E beany point 13

z E n O P>! Ci co [Q(to, Yo ) n rZ E E z- z < D r+l Then, for every k = 1, 2,..., we can select ck >, k >, zk E E, with ck O, k k 0 Zk + z as k k k, and Zk E CO [Q(t, y C) n (z c Er+l z i z i Also, for every k, we can select > 0 Zk, = 1,...,I, with p = r+2, such Y Y r+l i i that 0 < < 1, k, and Zk e [Q(t Yo Ck) n (z E E I Z Z I k y k ok r r l < k i = p...r], = 1... with zy = (Zkoy...zk ) c E r, and )' w k (z, k.,z),and z z7 k 1 k y k k7 k 1 2.... First we see that 1zk z i p,..,r, r = 1... k = 1, 2.... Then we see that for some k and k > k we certainly have Ek < 6, and hence k k' k for some (t Yk k) c MO namely (t k k E NE (to, 7 I 7) U 7 ) uc U(tk, ), 7 - 1,...,p, k >k. Here (t7 Y7) + (to, yo) as k +oo k k k - o k k' 7 = 1,...~. Since M is bounded we can select a subsequence, say still [k] 0 for the sake of simplicity, such that as k 0, we have uk + u, y 1, -,. Then Zk - zY as k y uy) with z = (t, ) with Iz7 - z I= O, or z = i z, i = p,...,r, r = 1,...,p. We can select the subsequence in such a way 0 7 that also k -\ as k -, O < k7, AY = 1. k Y Note that (tk 7M' uk) + (t y, u ) as k + 0, y = 1,...,., and property (U) of U(t, y) at (to, yo) yields u7 E U(to, yo), 7 = 1,...,p. Thus z = f(t, y0, u') c Q(t, y_), 7 = 1,...,, and also

z = z? 7 7 o p-l p r z (Z* z Z Z), W = z,...,. By o-convexity of the set Q(t, y ) we conclude that z Q(t, y). Statement (3.i) is thereby proved. 4. THE UPPER SET PROPERTY Let C, A(t), A be as in the first paragraph of 9 2. For every point r+l (t, y) c A, let Q(t, y) be a subset of E, r > O, We say that the sets Q(t, y) have the upper set property on A provided, for every (t, y) e A and 0 1 r for every point z - (z, z,...,z ) c Q(t, y), then any other point z = -0 1 r r+l -0 0 (-O 1 zr) Erl with z > z, is also a point of Q(t, y). The proofs of the following statements are not difficult and will be omitted. r+l (4.i) If S is a collection of subsets of E each of which has the upper set property, then the union of the sets in S has the upper set property. The closure and the convex hull of a set with the upper set property each have the upper set property. (4.ii) Let Q(t, y) have the upper set property on A. Let J(t), t c C, be -1 a measurable function on C which satisfies the inequality 0 < K < J(t) < K for some constant K and all points t c C. Then the sets QJ(t, y) defined by QJ(t, y) = (z c E I z = q ~ J(t) for q e Q(t, y)J, 15

have the upper set property on A. (4.iiji) Let Q(t, y) have the upper set property on A. Suppose that v(t) is a real valued function defined for t C C. Then the sets Q;(t, y) defined by ~~+ ~r+l Q+(t, y) (z E E z = (4(t),O,...,O) + q for q ~ Q(t, y)) have the upper set property on A. (4.iv) If the sets Q(t, y) have the upper set property and property (U), then they also have property Q(1). r+1 Proof. Let (t, y) be any point in A and z any point in E. Let 0 o 0 I and D be the sets I l fo n cl co (Q(to, yo, C) n N (z; r)) D = n cl (Q(t y, ) nN (z: r)) Property (U) holds provided D c Q(t, y0) n NP(zo; r)) and property Q(l) holds provided I C Q(to, yo) n N(zo; r). Therefore, we need only show that I c D. Let z be any point not in D with (z =(z0, z,...,z ). Then 16

z / ci (to yo, Eo) nN (z; r) for some E and f both positive. We denote by - the operation of projection o o 0 of E onto the z -axis of E. We have then -0o z To[cl(Q(t, Y0 E ) n N (z; r))] Since Q(t, y, e ) n N (z; r) has the upper set property, 0 z / cl co (Q(to, yo, ) N (z; r)). 0 We have z / I and may conclude that I c D. 5. EXAMPLES Examples of variable sets in E +1 with a property Q(p) for some 0 < p < r+l, but which do not have property (Q), or Q(r+l) arise naturally in control theory. Example 1. Let us consider the problem of the minimum of the cost functional 2 2 2 2 2 I[x, U U2 v],u2, v] ff ( x + + u2)d~dT with differential equations xy - ul, xt - u2, a.e. in G, and boundary condition yx - v s-a.e. on r = as, 17

where G = (, ) + 12 < i], r = aG is the boundary of G, s is the arc length on F, 7x the boundary values of x, and the control functions ul, u2, v have their values (yi 1) C U = E, v c V = (-l) U (1). Actually, we want to 1 2 minimize I in the class Q of all systems (x, ul, u2, v) with x any element of the Sobolev space W2(G) satisfying all relations and constraints above and for which I is finite. As we shall see in [8] the following sets are relevant: ~;fV ): o 1 2 0 2o 2 2 2 2 1 X) (Z Yz ) z+ + + 1 + a2 Z1 2 2 5 z =u2, (u, u ) e E] CE3: ~ 0 0 2 R [(z > z v, zv E For the sets Q we have r = 2, and they have property (Q), or Q(3), in A - cl G x E1. For the sets R we have r = 1, and they have property Q(1) in B - r, have property (U), but they are not convex, and do not have property (Q). We shall see in [9] that the problem above has an absolute minimum in Example 2. Let us consider the problem of the minimum of the cost functional I[x, Ul, u v] ff (x + x + x + x u + u2(1 - u2) ) dfd + fr(yx -1)2ds, with differential equations x + x =L u x, +x x u2 a.e. in G, ~~ rrnl ~ ~ 2

x: - cos v, x = sin v, s-a.e. on r = 2G, where G and r are as in example 1, where yx denotes the boundary values of x, and the control functions ul, u2, v have their values (U1, u2) c U = E2, v c V E1. We want to minimize I in a class Q of systems (x, Ul, u2, v) with x any element of the Sobolev space W2 (G) satisfying all relations and constraints above, for which I is finite, and satisfying an inequality ||il2 +fx 11 2 + Ifx 11 < M. Here M is a constant chosen large enough so that P is not empty. As we shall see in [8] and [9] the following sets are relevant: o 1 2 o 2 2 2 2 2 Q(y) [(Z z z) z > y + y2 + y + + u2 (1 - u2 z>2 *Y3 *z 2 2 1 2 z _ Uly Z= u2, (ul, 12) c E2] C E3, o0 1 2)1 2 1 2 ~R() = (>z os v,z - sin v, v C E1] C E, where y = (Y!, Y2, Y3) in Q(y), and y in R(y) are arbitrary. For both sets we have r - 2. The sets Q have property Q(2), but they are not convex and do not have property (Q), or Q(3). The sets R have property Q(1), but they are not convex and do not have property (Q), or Q(3). They all have property Q(o), or (U). We shall see in [9] that this problem has an absolute minimum in Q. 19

References [3] A. F. Filippov, On certain questions in the theory of optimal control. SIAM Journal on Control, Vol. 1, 1962, pp. 76-84. [2] L. Cesari, Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. I and II. Transactions of the American Mathematical Society, Vol. 124, 1966, pp. 369- 412, 4-13- )429. [3] L. Cesari, Existence theorems for multidimensional Lagrange problems. Journal of Optimization Theory and Applications, Vol. 1, 1967, pp. 87-112. [ 4] L. Cesari, Seminormality and upper seminormality in optimal control. Journal of Optimization Theory and Applications, Vol. 6, 1970, pp. 11413r7. [5] A. Lasota and C. Olech, On Cesari's semicontinuity condition for set valued mappings. Bulletin de l'Academic Polonaise de Science, Vol. 16, 1968, pp. 711-716, [6] C. Olech, Existence theorems for optimal problems with vector valued cost functions. Transactions of the American Mathematical Society, Vol. 1C6, 1969, pp. 157-180. [7] C. Olech, Existence theorems for optimal control problems involving multiple integrals. Journal of Differential Equations, Vol. 6, 1969, pp. 512-526. [8] D. E. Cowles, Lower closure theorems for Lagrange problems of optimization problems with distributed and boundary controls, Journal of 20

Optimization Theory and Applications. To appear. 39] L. Cesari and D. E. Cowles. Existence theorems for optimization problems with distributed and boundary controls. Archive for Rational Mechanics and Analysis. To appear. 21

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