TH E U N IV E R S I T Y OF M I C H I G A N COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report No. 17 EXISTENCE THEOREMS IN fJIULTIDIMENSIONAL PROBLEMS OF OTPTIMIZATION WITH DISTRIBUTED AND BOUNDARY CONTROLS Lamberto Cesari David E. Cowles ORA Project 02416 submitted for: UNIT'ED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOSR-69-1662 ARLINGTON, VIRGINIA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR February 1971

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EXISTENCE THEOREMS IN MULTIDIMENSIONAL PROBLEMS OF OPTIMIZATION WITH DISTRIBUTED AND BOUNDARY CONTROLS* Lamberto Cesari and David E. Cowles 1. INTRODUCTION In the present paper, we consider multidimensional nonlinear problems of optimization of the Lagrange type involving a cost functional expressed by means of integrals on a fixed domain G in Euclidian space Ev, v - 1, and on its boundary, aG, and also involving state equations-usually partial differential equations-in G and on aG, and controls both in G (distributed controls), and on aG (boundary controls), while our state variable x is an element of a Banach space S. The state equations, both in G and on aG, are written in abstract functional analysis form, and hence may represent partial differential equations or more general functional relations. The state equations, both in G and on aG, may be written in the so-called "strong" form, or in the'"weak'" form, as usual in partial differential equations theory. This paper extends to the present situation the method and ideas of previous papers by Cesari [3abc], and particularly of [3e]. Let G be a fixed bounded open set in E, v > 1, and let r be a given closed subset of aG on which we have a hyperarea measure AL. Let S be a Banach space of elements x, and let 2, I, J be operator on S, not necessarily linear, with values in the following spaces: *Work done in the frame of US-AFOSR Research Project 69-1662 at The University of Michigan. 1

~':S + (Li(G)),' S + (L~())r, f: S + (L1(G))s': S + (L(F))' where r,s,r' and s' are given positive integers. For every t = (t,...,t ) in the closure of G. let A(t) be a nonempty closed subset of the y-space Es. Let A be the set of all (t,y) such that t 1 s E cl(G) and y = (y,...,y ) c A(t). For every (t,y) c A, let U(t,y) be a nonempty subset of the u-space Em, u = (u,...,u ). We define analogous sets on a closed subset r of aG as follows. For every t e r, let B(t) be a nonempty closed subset of the y-space E, = (.,y. Let B be the set of all (t,y) with t E r and y E B(t). For every (t,y) E B, let V(t,y) be a m 1 m nonempty subset of the v'-space E, v = (v,...,v ). We consider the problem of finding an element x of S, a measurable con1 m trol vector u(t) = (u,...,u ), t c G, and a [-measurable control vector v(t) (v,...,v ), t c r, so as to minimize the cost functional: I[x,u,v] = f f (t,(~x)(t), u(t))dt + S go(t,( x)(t), v(t))dli G r subject to the state equations: Q~x)(t) = f(t,(?)(t), u(t)) a.e. in G, (1.2) (Jx)(t) = g(t,(~x)(t), v(t)) L-a.e. in r, (1.3) and the constraints: (77)(t) E A(t), u(t) e U(t, Cx)(t)) a.e. in G, (1.4) (Nx)(t) E B(t), v(t) E V(t,(Ax)(t)) [-a.e. in r. (1.5) 2

Here, u'(t) is said to be a distributed control, and v'(t) a boundary control. State equation (1.2) usually represents a system of partial differential equations, and (1.3) usually represents boundary data, or boundary controls, but may well represent a system of partial differential equations and related constraints and controls on the boundary aG of G. State ecuations (1.2) and (1.3) are said to be written in the strong form. We shall consider in ~ 6 also the problem of minimizing the cost functional (1.1) when (1.2) and (1.3) are written in the corresponding weak fOrm as is usual in the theory of partial differential equations. The corresponding results are framed in the present general theory with no extra effort. 2. PRELIMINARIES In order to state our lower closure and existence theorems, we will use C. B. Morrey's definition of a regular transformation of class K from his paper [8a]. Let X and Y be subsets of a Euclidian space E. A transformation x = x(y) of X onto Y is said to be of class K provided it is one to one and continuous, and the functions x = x(y) and y = y(x) satisfy uniform Lipschitz conditions on each compact subset of X and Y. respectively. In addition, the transformation is said to be regular if the functions x(y) and y(x) satisfy uniform Lipschitz conditions on the whole of X and Y, respectively. By a region, we mean an open connected set. If G is a region in EV, >G denotes its boundary and clG its closure. In stating our theorems, we will use properties of set valued functions...~~~~~~~~~~~~1

We now give the definitions of two well known properties of set valued functions. We shall use the notations of 1 I. Also, given a point (t,y ) E A and a number o > 0 we denote by N(to,yo) the set of all points (t,y) E A at a distance < 6 from (t,y ). r+l For every (t,y) E A let Q(t,y) be a subset of the z-space E, z = 0 r (z,...,z ). We say that the sets Q(t,y) have Kuratowski's upper semicontinuity property [6], or property (U), at the point (t,y ) C A provided Q(to,yo) = n Cl Q(t,yo,), where Q(to,Yo ) - U Q(t,y) (t,y) N (t,yo) We say that the sets Q(t,y) have Cesari's upper semicontinuity property [3a], or property (Q), at (to,y ) E A, provided Q(t,yo) = n cl co Q(to,yo,C). We say that the sets Q(t,y) have property (U), or (Q), in A if they have this property at every point (t,yo) E A. Sets having property (U) are closed, and sets having property (Q) are closed and convex. It was found useful to introduce intermediate properties Q(p), 0 < p < r+l, of variable sets (D. E. Cowles [4a]). Let p be any integer, O < p < r+l. We say that the sets Q(t,y) have property Q(p) at the point (t,y ) E A provided for every z = (z,z,...l r+l r+l' z )EE, o~~~~~~~~~~~~

o r ) ri i Q(to,y) (7= (z,...,z z i... O. For p = r+l we understand that the sets in braces in the first and second members of this relation coincide with E. Also note that if the sets Q(t,y) have property Q(p) at the point (t,y ) E A, then for every z = (z,z,... r+l r+-1 z ) E the set Q(ty )y) n(z E Iz = z, i is closed and convex as intersection of sets having the same property. Thus, o p-1 P r (z. 0 0 sin' tis riz, ) o, Q(ty), (2.i) For anyinteger p, 0 < p r, property Q(p+l) implies property Q(p). Also, property Q(r+l) holds if and only if property () holds, and property Foran t vro point (ty) E A, let Q(ty) be a subset of e We say that the sets (ty) have the "upper set property" on A provided (ty), For a proof of this statement see (D. E. Cowles [4a]). -::

o 1 r 0O1 r f withr o0 A, z = (z,z,...,z ) Q(t,y), o = (oz,. Z) Er+l o o plies z c Q(t,y). (2.ii) If the sets Q(t,y) have the upper set property on A and also the property (U), then the same sets have property Q(1) on A. For the proof of this statement see (D. E. Cowles [4a]). 3. A LOWER CLOSURE THEOREM Let G be a bounded measurable subset of EV, v > 1, whose boundary will be denoted by aG. Let Fr, j = 1,...,N, be subsets of aG, each of which is the image under ~~~~J -~~~~~~~~~V1 a regular transformation t. of class K of a bounded interval R' of E. Let J 3J N N P' be a. closed subset of Uj1 ry and let be a measure defined on U. 1'r.. ~ ~~ ~ ~~~~jl jj=l For each j = 1,...,N, we assume that if e is a subset of r., measurable with respect to Ai, then E = t. (e) is measurable with respect to Lebesgue (v-1)dimensional measure I I on R'. Also, we assume that the converse is true, so that measurable sets on r. and R' correspond under t, j =...N. J jo J Finally, we assume that there is a constant K > 1 such that if e = t.(E) is p-measurable, then K- ZEl < (e) < KIE, (3.1) independently of j = 1,...,N, and e. Since p. induces a measure on each set R' via the transformation tj, j = 1,...,N, we may define Jj (T), T C R, as d d the function in LI(R') which satisfy the relations 6

~(t (E)) = I J.(r)dr (3.2) for every measurable subset E of R'., j 1,...,N. For every t E clG, let A(t) be a nonempty closed subset of the y-space E, y = (y,...,y ). Let A be the set of all points (t,y) with t c clG and y c A(t). For every (t,y) E A let U(t,y) be a nonempty subset of the u-space Em 1 m s m E, u = (u,...,u ). Let M be the set of all (t,y,u) c E E x Ex such that (t,y) c A and u c U(t,y). For every t c r, let B(t) be a nonempty closed subset of the y-space E (l = (y,..,y). Let B be the set of all (t,y) with t E r and e B(t). For every (t,y) E B, let V(t,y) be a nonempty closed subset of the v-space Em' 1 v' s v = (v,...,v ). Let M be the set of all (t,y,v) E E x E x E with (t,y) E B and v E V(t,y). Let f(t,y,u) = (f,f) = (f,fl,..,f ) be a continuous (r+l)-vector o 01 r function on M, and for every (t,y) E A let Q(t,y) denote the set;(~ o o) ( 1 zr Er+l Q(t:,y) = z= (z0,z) = (,z,...1zr) E r+l z > f (t,y,u), z = f(t,y,u), u c U(t,y)]. Let g(tg,v) = (go,g) = (og,...,gr) be a continuous (r'+l)-vector 0 function on M, and for every (t,y) E B, let R(t,y) denote the set o) o 10 r''+l R(t) = (z z) = (zO,z l-.. z ) ) Er > g (t,,v), z = g(t,,v), v V(t,)). 7

We consider here the functional I[y,),u,v] = f (t,y(t),u(t))dt + f go(t,y(t),v(t))d~. (5.5) G r In the lower closure theorem below we shall deal with sequences of functions all defined on G and r: r1 r z(t) = (z (t),z ), k)' k t ) k y(t) (y,... k(t) = (Y,.'.,Yk) u (t) = (Uk..uk) t E Gj k = 1,2,..., k ko' k (t z ), zk(t) = (... ), y(t) = (,, Y ) (t) = (YYk 0 ) v (t) = (v1 mk(t) = (Vk,....,k )(3 (3.i) Theorem (a lower closure theorem). Let G be bounded and measurable, A, B, M, M closed, fo(t,yu), f(t,y,u) = (fl,...,f) continuous on M, Or 0 g(t,y,v), g(t,,v) = (gl,..,gr,) continuous on M, and assume that for some integers p, p', O < p < r, O < p' < r', the sets Q(t,y) have property Q(p+l) on A and the sets R(t,y) have property Q(p'+l) on B. Let us assume that there are functions *(t) > O, t E G, * E L (G) and 1(t) > O, t r, tE Ll(r), such that f (t,yu) > -*(t) for all (t,y,u) E M, and go(t,y,v) > -V(t) for 0 0 0 all (t,y,v) E M. Let us assume that the functions z (t), zk(t), yj(t), Yk(t), i =...,r, j =,...,s, are in L (G), that the functions uk(t) are measurable on G, 1 = 1,...,m, that f (t,yk(t),u (t)) c L1(G), and that 8

Yk t) ~ A t), Ukt) C U~t,ykvt) J Zkt) fi(t,yk(t),uk(t)) a.e. on G, k = 1,2,.... (3.5) i k k Let us assume that the functions z (t), zk(t), y(t), yk(t)) i k' k l,..,r', j = l,...,s', are in L1 (), that the functions v (t) are measurable in r, = 1,...,m', that go(t,yk(t),v (t)) E L (r), and that Bk(t):E B(t), vk(t) ~ V(t,~k(t)), k(t) = ((t),vk(t),v(t)) t-a.e. on r, k = 1,2,.... (3.6) i k k Finally, let us assume that as k oo we have i i z'(t) - z (t) strongly inL (G), i = p+l,...,r, k 1 yk(t) + yJ(t) strongly in L (G), j = 1,.,s, zk(t) i z (t) weakly in Ll(r), i = l,...,p', k zk(t) +z (t) strongly in Ll(r), i = p'+l,..,r', yk(t) - yi(t) strongly in L (r), j = 1. limk I[Yk,'kUkVk] < a < +o. Then, y(t) c A(t) a.e. on G, y(t) E B(t) i-a.e. on r, and there are measur1 m able functions u(t) = (u,...,u ), t E G, and i-measurable functions v(t) = (v,...,v ), t r7, such that f (t,y(t),u(t)) c L1(G), g(ty(t),(t))) oL(r), and such that 1 9'

u(t) c U(t,y(t)),z (t) ft)), i = l,...,r, a.e. on G, v(t) c V)tY(t)),Z (t) = gi(t,y(t),v(t)), i = l,...,r', r -a.e. on r, I[y,',u,v] < a 0 For the proof of this lower closure theorem see (L. Cesari [3e], D. E. Cowles [4b]). Remark. In applications, it often occurs that the sets U and V are fixed and compact, or alternatively U(t,y), V(t,y) are compact, equibounded, and have property (U) in A and B, respectively. Having assumed f, f, g, g 0 o continuous, then the sets Q(t,y), R(t,y) certainly are compact and have property (U) in A and B, respectively. If convex, then they also have property (Q) (see [3a]). If p-convex, they have property Q(p) (see [4a]). 4. AN EXISTENCE THEOREM FOR OPTIMIZATION PROBLEMS WITH STATE EQUATIONS IN THE STRONG FORM In this section we should mainly use the same notations of 1 3. For the sake of simplicity we shall denote by T the family of all measurable m-vector 1 mi functions u(t) = (u,...,u ), t E G, and by T the family of all CI-measurable 1 m m'-vector functions v(t) = (v,...,v ), t C r. Let S be a Banach space of elements x with norm ilxll, and let ~ A, c ) be operators, not necessarily linear, as described in S 1, or t: S-(L (G))r, S: + (Li(G)), / S + (Ll(I))r' {: S + (LL(r))si. We shall discuss here the problem of optimization (1.1-5) of ~ 1. 10

A triple is said to be admissible (for the problem (1.1-5)) provided x c S. u c T, V c Ty f (t,(7x)(t),u(t)) c L1(G), go(t,(~x)(t),v(t)) c Ll(r), and relations (1.2-5) hold. A class Q of admissible triples is said to be closed if the following occurs: if (xk,uk,vk) Q 1 k =,,..., Xk x weakly in S as k oo, limk I[x k,k,Vk] < a < +, and there are admissible triples (x,u,v) such that I[x,u,v] < a, then there is also some triple (x,u,v) cE, with I[x,u,v] < a. For a class R of admissible triples we denote by (X) the subset of S 0 defined by [x]) = (x E S|(x,u,v) c Q for some u E T, v E T). Note that for x c S, then z(t) = (z,...,z ) = (x)(t) c (L (G))r and we shall denote by i ith zi(t) = x) (t), t c G, the i component ofIx. Analogously, y(t) = 1) r (y,...,y ) = ()(t) E (L1(G)) and we set yj(t) = (- )J(t), t c G, j = 1,...,s; z(t) = (l,.,z ) = (Jx)(t) c (Ll(F))r and we set z (t) = (Jx) (t), t c r, i = l,...,r; (t) = )(t) (() and we set J(t) = (*)J(t), t E r, j = 1,...,s'. (4.i) Existence Theorem. Let G be bounded and measurable, A, B, M, M closed, f (t,y,u), f(t,y,u) = (fl,...,f ) continuous on M, go(t,y,v), 0 g(t,y,v) = (gl',..',g) continuous on M, and assume that, for given integers p, pt, O K p < r, <0 p' K r', the sets Q(t,y) have property Q(p+l) on A, and the sets R(t,y) have property Q(p'+l) on B. Let us assume that there are functions v(t) > 0, t c G, E c L (G), and *(t) > O, t c r, * E L(rF), such that f (t,y,u) > -~(t) for all (t,y,u) c M, and g (t,y,v) > -$(t) for all (t,y,v) E M. Let us assume that, for every sequence x,Xk, k = 1,2,..., of 11

elements of S with x + x weakly in S it occurs that there is some subsek quence [k ] such that gx )' - IAx)i weakly in L1(G), i = l,...,p, P k'x ) + ~x) strongly in Ll(G), i = p+l,...,r, P ( k )j + (~x)J strongly in L(G), j,... (Jx )'-+ (&x) weakly in L (F), i = 1,...,O, k 1 p V(cfX )' (Jx) strongly in L (r), i = p'+l,,r' (xk )J -+ (Kx)j strongly in L (r), j = 1,...,s',.P~ as p +. Let Q be a nonempty closed class of admissible triples (x,u,v) such that the set (x) is weakly sequentially relatively compact. Then,the functional (1.1), or I[x,u,v], has an absolute minimum in P. In view of statements (2.i) and (2.ii), note that, for p = r we actually require above that the sets Q(t,y) have property (Q), and for p = 0 we actually require that the sets Q(t,y) have property (U). Analogously, for p' = r' we actually require that the sets R(t,y) have property (Q), for p' = 0 we require that the sets R(t,j) have property (U). In general, for 0 < p < r, 0 < p' < r', properties Q(p+l) and Q(p'+l) represent intermediate requirements. Proof. Let i be the infimum of I[x,u,v] in the class R. Then i is finite, and we consider a minimizing sequence for I in 9, that is, a sequence 12

(xk,u k,v k), k = 1,2,..., of admissible triples, all in Q, with I[xk,uk,vk] I i as k ~+ c. Since the set (x} is weakly sequentially relatively compact, there is some element x E S and some subsequence of [xk] which is weakly convergent to x. For the sake of simplicity we denote such a sequence by fk], and thus xk + x weakly in S. As a consequence, there is a subsequence [k ] p for which convergence relations (4.1) hold. We shall denote this subsequence again by [k]. By using the notations zk A k)() Yk(t) = (xk)t) (t), (t) = x)(t), y(t) = )(t), t ~ G, ek(t) = (X k)(t), yk(t) = (k)(t), (t) = (Jx)(t) N(t) = (x)(t), t E r, we see that relations (1.2-5) imply yk(t) E A(t), uk(t) c U(t,yk(t)), zk(t) = fi(t,yk(t),uk(t)) a.e. in G, i =,...r, yk(t) c B(t), vk(t) E V(t,k(t)), zk(t) gi(t, k(t),vk(t))-ae in, i =,r' k i i i i In addition, zk 7 z weakly in L1(G), i = l,...,p, zk z strongly in L1(G), i 02oi 0i i = p+l,..., r, Yk +y strongly in L1(G), j = 1,...,S, Zk z weakly in ci o. I j - oj Ll(r), i = l,...,p', z -. z strongly in Ll('), i = p'+l,...,r', + Y 13

strongly in Ll(F), j = 1,...,s'. Finally,the sets Q(t,y) satisfy property Q(p+l) on A, and the sets R(t,y) satisfy property Q,(p'+l) on B. We can now apply lower closure theorem (3.i). Then, y(t) = (4x)(t) E A(t) a.e. on G, y(t) = (x)(t) c B(t) Pi-a.e. on r, and there are elements u c T and v c T such that i u(t) c U(t,y(t)), z (t) = f.(t,y(t),u(t)), i = l,...,r, a.e. in G, g.(t,y(t),v(t)), i = l,...,r', p-a.e. on F, o.1. o -a e. on O (t,y (t),u(t)) C L(G), go(t,(t),v(t)). Ll(F), I[x,u,v] _<, (4.2) that is, the triple (x,u,v) is admissible. Since P. is a closed class of admissible triples, there is in Q some admissible triples (x,u,v) with I[x,u,v] < i, and relations (4.2) hold also for u,v replaced by u,v. Since i is the infimum of I in X, we have I[x,u,v] > i, and finally I[x,u,v] = i. Remark i. Theorem (4.i) holds even if we replace the cost functional (1.1) by another analogous one with an added term J[x], provided we know that J[x] is lower semicontinuous functional on S with respect to weak convergence on S. That is, we need only require on J that xk + x weakly in S implies limk J[xk] > J[x]. 14

Remark 2. In each of the theorems (3.i), (4.i), (4.ii) we could have assumed that G and r are each made up of a finite number of components on each of which there is a distinct system of state equations. Remark 3. Of particular interest is the case where the element x of an admissible triple (x,u,v) uniquely determines the controls u and v. That is, (x,u,v), (x,u,v) admissible implies u = u a.e. on G, and v = v Ct-a.e. on r. In this situation, lower closure theorem (3.i) reduces to a lower semicontinuity theorem, and corresponding particular existence theorems could have been obtained by standard lower semicontinuity argument. This holds, for instance, for free problems of the calculus of variations and other problems. Remark 4. As mentioned in the remark at the end of 5 3, if U and V are fixed compact sets (or U(t,y), V(t,y) are compact, equibounded, and have property (U)), then the sets Q(t,y), R(t,y) certainly have property (U) and, if convex, have property (Q) (see [3a]). Also, an analogous statement holds for the intermediate properties Q(p) in the sense that, if p-convex, than they have property Q(p) (see [4a]). Example 1. The following example, mentioned by Fichera [5], illustrates existence theorem (4.i). In this example the particular situation depicted in Remark 3 occurs, and therefore our lower closure theorem (3.i) reduces to a lower semicontinuity theorem (which includes Fichera's lower semicontinuity theorem). Let G be a bounded open subset of EV of class K, v > 1 (see 2). Let I be the hyperarea measure defined on the boundary r = aG of G. Let 15

W (G) be a Sobolev space on G for real p, 1 < p < + 00 and integer 1, 1 < I + o, and usual norm IlXII = Ilxii Q = z~ (JI 1/(t)l PdtP W (G) IC<I G P where D x denotes the generalized partial derivative of x in G of order a = ( 1...,a ), and a = al+...+a. Let X- be the linear subset of W (G) V v o p made up of all functions x which are continuous on G U 6G, together with their partial derivatives of all orders. For t c aG let ((x)(t) denote the vector (4'x)(t) = (7xYx,x,...,7x) where -f denotes the boundary values of f and Vx, 0 < j < h-l, denotes the vector of all partial derivatives D x of order |al = j. Let s' denote the total number of components of the vector (XJx)(t). For given real valued functions a (t), t E aG, with al= i, a E L (6G), let (Jx)(t) denote the real valued function (2x)(t) = Z a (t) D x(t), t E aG. I -al=m Finally, let S be the completion of X with respect to the norm 0 Ilxll = lxlli2 + IaIJxll2 1/2 W (G) L (G) 16

elude that Jx also is defined on S. We are concerned with the problem of the minimum in S of the functional I[x] = f g (t,(4x)(t),(Jx)(t))d~, (4.3) aG where g is a given continuous function on the closed set M = G x E x E'. O This problem is immediately reducible to the form (1.1-5) by taking fx =! 0, 7x = O, f = O, by taking B(t) = E for every t C r = aG, and V(t,y) = E' 0 S for every (t,y) C B = aG x E. Thus, there are no constraints on the control variable v, or v C V = E'. We have now the problem of minimum of the type (1.1-5) with functional I[x,v] = J go(t,(4x)(t),v(t))d~ F and state equations (on the boundary) (Ox)(t) = v(t), a.e. on r= 3G. In the present situation, and using the notations of 3 and 4, we have r = s = m = O, s' as above, r' = m' = 1. The sets R are the sets R(t, ) = = (zz) E E2z > g (tj,v), z = v c E ) ((z,v) C E Iz > g(t,y,v), v C EJ, and thus they are convex if and only if g (t,y,v) is convex in v for every (t,y) E B = axG x E. Finally, if we assume that there is some real-valued continuous function 17

)() O < S < to, with g (t,y,v) > O(jv ) for all (t,y,v) E M, ~(~)/. I + + o as ~ + o+ (4.4) then we know from (Cesari [3b]) that the sets R(t,y) satisfy property (Q) on B. Note that, if p = Min O(0), < ~ < +0, then relation g (t,y,v) > -{(t) holds for all (t,y,v) C M with f = -f.JI, a constant. If p > 1, then for any sequence of elements x, xk, k = 1,2..., of S with xk + x weakly in S as k + o, certainly there is a subsequence [kp] such p that Jxk + Jx weakly in L (aG) and XXk + - x strongly in (L (G))S as k 0. p. p p P P We may take y(t) = (0x)(t), Yk(t) = (Xk)(t), t E G, and z(t) = v(t) = (x)(t), zk(t) = vk(t) = (Cxk)(t), t ~ aG, k = 1,2,.... By lower closure theorem (3.i) with p = r = 0, p' = r' = 1, we derive now the following lower semicontinuity theorem concerning the integral I[x]: 0 5 (a) If g (t,y,v) is continuous on M = aG x E x E' and convex in v for every (t,y) c B = aG x Es, if there is a function (T), 0 < ~ < + oo, such that (4.4) holds, then the functional I[x] is lower semicontinuous in S; precisely, x, Xk ~ S. xk + x weakly in S as k + o, lim I[xk]I< < a < + implies I[x] < a. Growth condition (4.4) can be disregarded if we know that the sets R(t,y) satisfy property (Q) on B, and that go(t,y,v) > - (t) for all 18

(t,~,v) c and S'ome f > 0, > L? (3G) Also, note that under the assumed hypotheses go(t, x)(t),(J(x))(t)) is certainly p.-measurable on 6G and >-r(t) with $ c L (G) Thus, the functional I[x], or (4.5), is always defined in S, either finite, or + co. Let ON be the class of all x c S with xIItxlJ < N and I[x] finite. If i N denotes the infimum of I[x] in PN, then, by force of go > -, i is finite, 0. and in the search of the minimum of I[x] in P., we can restrict ourselves to the subclass O, of all x E DN with I[x] < i+l. Any such class N' is obviN NN ously weakly compact in the topology of S, and weakly closed in the same topology. From (4.i)we may now derive the following existence statement: (b) Under the conditions of (a) with 1 < p + oo, the functional I[x] has an absolute minimum in any nonempty class'N Example 2. Let G be a connected bounded open subset of the ~T-plane E 2 If G is of class K, then the usual are length measure s is defined On r = cG. We are concerned with the minimum of the functional I[x,u,v] = 7 ff (f, ),x Cx xu)d d dB 0 ~ G + f g (S,r)X,7x,7x,x,v)ds,, (4.5) 6Go with state equations 19

x ~r + x - fI= t(~,x,xrx,u) a.e. in G, a( 0,y)x + b(,v)yxx + c(,T x = g(5,rj,yx,7yx,yx,v) s-a.e. on aG (4.6) and constraints u(5,s) E U(,I,x,x),x) a.e. in G, v(5,n) c V(T,T,yx,yx,Yxn) s-a.e. on aG. (4-7) Here yf denotes the boundary values of f. This problem is immediately written in the form (1.1-5) by taking xx = x + X?T(x = (xxx), Jx = ayx + byx~ + cyxB, ~x = (yx,yxS,yx ), r = r' = 1, s = s' = 3. We take for S the Sobolev space S = W2(G), p > i, p and we assume that the given functions a, b, c are of class L (G). Note that (x)(%,) could be the normal derivative of x at ( c,) E aG if only a = 0, and b, c the direction cosines of the normal to aG at (e,TI)(s-a.e. on aG). Also, we take A(%,TI) = E5, B(5,) = B3, hence A = (clG) x E3, B = (HG) x E3. For the sake of simplicity we assume m = m' = l, so that, if y = (y,y,y3) E E3, y = (y,y,y3) E E, then U(%,9,y) denotes a subset of E for every (~,i,y) E A and V(,rl,r) a subset of E for every (%,~~) e B. 20

Finally, if M, M are the corresponding sets, Mc(clG) x E x E, MC(6G) x E3 x E1, then f, f are rea-l valued continuous functions on M, and g o g 0~~O0 are real valued continuous functions on M. We consider here the sets Q(5,T,y) = ((z,z) c E Iz > f_(, yyu) z f(=,,yu), u c U(,, (,,) = (,z) E > g_(,,yv) z = g( y), v c } for every (y,r,y) E A and for every (C,qy) E B. respectively. Note that, if xk + x weakly in S = W (G), then there is certainly a sub-.k p sequence [k ] such that Axkk +x weakly in L (G), x k + lx strongly in P k p P (L (G))5, fxk + Jx strongly in LL(6G), txk +- x strongly in L (3G). An p k~ p P admissible triple is now a triple (x,u,v) with x c W (G), u measurable on G, P v s-measura.ble on aG, satisfying (4.6), (4.7), and such that f (',y,x,xx,u 0 c L (G) and go(,,y x,yx,yx,v) c Ll(G). We can now derive from (4.i) with p = r, p' = 0 the following existence statement: (c) Let G be connected bounded open and of class K in E, let M, M be closed, let f, f be real valued and continuous on 0 M, and go, g real valued and continuous on M, and assume that the sets Q(~,r,y) satisfy property (Q)on A = (clG) x 21

E) and the sets R(~,y,j) satisfy property (U) on B = (~G) x E3. Let us assume that there are functions 0(~,r) > 0, tC L(G) and t(~,) > O, $ C Li(aG) such that f (Lr,y,u) > -V(t,n) and go(S,~,Vy) > -K(5,r) for all (~,r,y,u) C M and ( 0,r,y,v) C M, respectively. Let D be any nonempty class of admissible triples (x,u,v) for which the set (xli is norm bounded in W (G), p > 1; that is, there is a constant N such that P (x,uv) C X+ implies lxii + 1lx || + ix || < N. Then, the cost functional (4.5) has an absolute minimum in P. Remark 4. Many examples of optimization problems with distributed and boundary controls and state equations in the strong form are of the same general form of example 2 above. The equations (Vx)(t) = f(t,(9)(t),u(t)), t e G, is a partial differential equation (or a system), and the equation (Jx)(t) = g(t,("x)(t),v(t)), t E aG, represents a certain set of constraints on the boundary values of the state variables. The conditions of theorems (4.i) are usually satisfied with p = r and p' = 0, that is, we require property (Q) on the sets Q and property (U) on the sets R. One more example is in ([4a], no. 5, example 1). In applications it often occurs that U and V are fixed compact sets (or alternatively U(t,y), V(t,y) are compact equibounded variable sets satisfying property (U)). In either situation, the sets Q(t,y), R(t,y) certainly have property (U) and, if convex, have property (Q). (Analogously, if p-convex, then they have property Q(p) [4a].) A number of problems with U, V fixed 22

compact sets have been considered in [3h], and existence theorem (4.i) may easily be applied. Example 3. In this example we wish to illustrate the use of the intermediate properties Q(p). Let us consider the problem of the minimum of the cost functional 2 2 2 2 2 2 2 I[x,ul,u2,v] = ff(x +xx + +u +u2(1-u) )d~d + f(7x-1) ds, G r with differential equations x + = x + = u, a.e. in G, Yx: = cos v, yx sin v, s-a.e. on r = aG, where G = [(r,q)J + q < 1], r is the boundary of G. where yx, yx, x denote the boundary values of x, x:, x, and the control functions ul, u2, v have their values (ul,U2) c U =E2 E, V = E1. We want to minimize I in a class P of systems (x,ul,u2,v) with ul, u2 measurable in G, v measurable on r, x any element of the Sobolev class W (G) satisfying all relations above, 2 satisfying an inequality Jjxj j + Jx 11 + Jx 11 < M, and for which I is finite. Here the constant M is assumed to be sufficiently large so that n is not empty. We may well consider only those elements of Q for which I < N for some constant N. Here we have f > 0, g > 0, and we can take O = 0, O = 0. 0 0 Also, we have, x = (x:: +x, x +x ),Rx = (x,xx ), Jx = (Yx,x), x = 7x, r = 2, s = 3, r' = 2, s' = 1, m = 2, m' = 1. Then, for any sequence [xk] of elements x ~ (x), certainly there is a subsequence, say still [k] for the 23

1 2:,akw-' Silr pLLoi Ly, wiLl x x wenllly ill S W (() f'ror' s0rle x c SI, atntid (Joxh) )l (a~x)1 nweeakly in L2 (G), (;xk) - (ax)2 strongly in L2(G),?x -*Ax strongly in (L2(G))3 k +Jx Jx strongly in (L2(F)) txk + x strongly in L2(r). We consider here the sets o12 o 2 2 2 2 2 2 z 1 )1z > 2+y2 +y + +u2(u), Q(y1, yy) = [(,z Z 2 2 2 2 1 2 z = U1, = u2' (U',U2) c E2] o 12 2 1 2 y ) [(z,z 1,1z )I > (y-l), z = cos v, z = sin v, v E E1]. The sets Q c E have property Q(2), the sets R c E3 have property Q(1) and all have property (U), or Q(O). They are not convex, and do not have, therefore, property (Q), or Q(3). Nevertheless, existence theorem (4.i) applies with p = 2, pt' 0, and the problem under consideration has an absolute minimum in P. 5. ANOTHER EXISTENCE THEOREM FOR OPTIMIZATION PROBLEM WITH STATE EQUATIONS IN THE STRONG FORM We now consider the case where the operators A, 1, A, t themselves depend on x C S and on suitable components of the controls, instead of depending on x alone as in ~ 4. Thus, the theorem we shall prove here is, for practical purposes, more general than theorem (4.i). Nevertheless, we shall prove it as a corollary of theorem (4.i). We shall consider here additional spaces of distributed and boundary controls,QandQ, with elements ~ c and ~ EYJ respectively, both(and Qbeing given Banach spaces. 24

m It may occur that O and O are vector functions (t) = (u...,u ), m'+l t c G. and v (t) = (v,...,v ), t c r, and in this case the control u = (u, ~) is an m-vector function on G, and v = (v, 0) is an m'-vector function on r. In any case, we write our controls as u = (u, ) and v = (v, 0). We are concerned here with the problem of the minimum of a functional I[x,u, ~,v, ] = f f (t,(R(x, ))(t),u(t))dt 0 G + f go(t,((x,~ ))(t),v(t))d (5.1) subject to the state equations ((x, M))(t) = f(t,(N(x, )(t),u(t)) a.e. in G, (5.2) (J(x, ))(t) = g(t,(Q(x, )(t),v(t)) [L-a.e. in r, (5.3) and the constraints ((x, I ))(t) E A(t), u(t) E U(t,(N(x, ))(t)) a.e. in G, (5.4) ({(xj ))(t) c B(t), v(t) E V(t,(f(x, ))(t)) t-a.e. in r. As in S 1, G is a fixed bounded open set in EV, v > 1, and r a given closed subset of aG on which we have a hyperarea measure t. Let S,Q j3 be Banach spaces of elements x,, ~, and let',, ) be operators on S x ~, S x Q, not necessarily linear, with values in the following spaces: 25

?,: Sx @ (L (G)): S x L (Ll(r)) where r, s, r', s' are given positive integers. Let A(t), A, U(t,y), M and B(t), B, V(t,y), M be the sets defined in S 1 and 5, M c (clG) x ES x Em M c r x E x E, U(ty) c E V(t) let f (tyu), f(tyu) (f f) be defined on M, and g (t,,v), gt, 0 yyyu= r y 0Er+lr g(t),v) = (gl...gr,) be defined on M. Let Q(t,y) c E be the sets defined in S 3 for every (t,y) c A, and let R(t,) c Er+l be the analogous sets also defined in 1 3 for every (t,y) c B. As in S 4, we denote by T the set of all measurable m-vector functions u(t) = (u:...,u ), t c G, and by T the set of all measurable m'-vector functions v(t) (v 1...,vm) t C r. A triple (x,u,v), or system (x,u, ~,v, ~), is said to be admissible (for the problem (5.1-5)) provided x c S, u c T, v c T, ~ c ), 0 c 0, f (t,(T(x, ))(t),u(t)) c L (G) g (t,(K(x, ))(t),v,v(t) c L (), and relations (5.2-5) hold. A class R of admissible systems is said to be closed if the following occurs: if (Xkuk,(,vk, ) C k = 1,2,..., xk + x weakly in S, u -+ ~ weakly in Q, v + weakly in Q as k + m, limk I[xkuk, ), Vk, V ] < a < + o, and there are admissible systems (x,u, ~,v, ~) such that I[x,u, 0,v, ]3 < a, then there are also systems (x,u,,v, ) c D with I[x,u,,v, ] < a. 26

For a class Q of admissible systems (x,u, ),v, v ) we denote by (x)], (5) ) ( 0 tile.;letl defiiicle as {x] in 4. (5.i) Existence Theorem. Let G be bounded and measurable, A, B, M, M closed, f (t,y,u), f(t,y,u)= (fl,...,f ) continuous on M, g (t,y,v), o 0 g(t,y,v) = (gl'...,gr,) continuous on M, and assume that for given integers p, p', 0 < p < r, 0 < p' < r', the sets Q(t,y) have property Q(p+l) on A, and the sets R(t,y) have property Q(p'+l) on B. Let us assume that there o o0 are functions x(t) > O, t c G, 4 c L (G), and *(t) > O, t c r, C L (r), such that f (t,y,u) > -*(t) for all (t,y,u) C M, and g (t,y,v) > -j(t) for all (t,y,v) c M. Let us assume that for every sequence x,,,xk, O, O, rk = 1,2,..., of elements, X,Xk C S, (), i C, ( with xk + x weakly in S, O + O weakly in Q, + O weakly in, it occurs that there is some subsequence [k ] such that (.(xk, u ))k + ((x, ))i weakly in L1(G), i = l,...,p, P k((Xk,n))i + (J(x,~))i strongly i = p+l,...,r, (N(Xk p ))u - (r(x,~))i strongly in L1(G), j = 1,...,s, p (i(xk, v )) (J(x, ))i strongly in L (r), i = l'...', (J(Xk v )) j + (J(x, )) strongly in Ll(f), i = p,...,s', as p. Let be a nstronglempty closed class of admissible systems (x,u, 27

V, ) such that (Ix}, ( 3, ( ~ ] are weakly sequentially relatively compact. Then, the functional (5.1), or I[x,u, Q,v, ], has an absolute minimum in P. Proof. Apply existence theorem (4.i) with S replaced by S x x and x replaced by (x, u, v ). Example (a problem of evolution in strong form). Let G be a subset of v~l 1 V the tT-space Ev 1, T = (T,.,T ),of the form G = (o,T) x G', where G' is an open bounded connected subset of Ev of class K. Thus, vfl replaces v, and r = 6G is made up of three parts Fl = (o] x G', = (o,T) x aG', 3 = (T) x G'. 1 2 3 On Pl and r3 we have the Lebesgue v-dimensional measure, or I I (and we shall use the symbol dT in integration). In F2 we have the product measure o = x t of the one-dimensional measure on [o,T] and of the hyperarea i =. 11 on the boundary 6G' of G' (and we shall use the symbol dt dt in integration). Givena function x in G, we shall denote by yx the boundary values of x on r = aG, and specifically we shall denote by y.x the boundary values of x on Fi, i = 1,2,3. We shall denote by T, T. the families of all measurable functions on G, Fi, i = 1,2,3, respectively. We shall denote by S (G), 1 < p < + c, I > 1, the space of all real valued functions x(t,T), (t,T) E G, such that ax/at and all D x, c = (1,..., a ), 0 < Ja < ~, exist as generalized derivatives, and are all in L (G). We _ P shall make S (G) a Banach space by means of the norm

I xll = JIxJl S (G) P [(G dtT 2/p +2j /2 = I~ ~x,/tPdtdr + -C Z.I X PdtdT These spaces S (G) have been studied by J. P. Aubin [1], who has proved for these Pi.' spaces weak compactness properties similar to those for the Sobolev spaces. Each element x c S (G) possesses boundary values y X, Y x on F respecP 1 3 tively, and all y2 DX, 0 < f c < ~-1 on r2 We are concerned with the minimum of a, functional I[xu,,v2, 2,v3] = f (t,%(~x)(tT),u(tT))dtdT G + f go(tT, (9x )(t,T),v(t,T))dtd9 2 + gO( (tx) (T,)T),3 (T) )dT ~0 3 with state equations (in the strong form) 4(x,' )(t,T) = f(t,T,( x)(t,T),u(t,T)) a.e. on G, J(XQ)(tT) = g(t,T,(,X)(t,T),v2(t,T) ) o-a.e on F2, and constraints (1x)(tT) c A(t,T), u(tT) E U(t,T,(Pqx)(t,T)) ae. on G, (Ox)(t,T) E B2(tT), v2(t,T) E V2X)(t,T)) o-a.e. on?2, (Q'x)(T,T) c B3(T), v3(T) c V3(T,()C)(T,T)) a.e. on %, 29

and no boundary conditions on fl (in other words, the initial values (T) x(o,T) are free and' (i), e FIl, is itself a control, therefore. 0 0 We take now x C S = S (G), u c T, v2 e T2, v c T3, and E, v2 E p 2 2 5 I, where O and are weakly closed subsets ofL (G) and L (f'), respectively, and both are norm bounded (in the norms of L (G) and L (r2)). p p 2 Above 2, t,, J are operators, not necessarily linear, say S + (L (G))S p S (L (ra)), X: S X (L (G)) c: S x i + (L(P2))r. We assume that xk x weakly in, S _ + ~ weakly in L (G), weakly in L (rP) implies that 1xk +7nx strongly in (L (G)) s, tk - x strongly in (L (P)), J(xku) + (x,~) weakly in *(L (p )), J(x, p k' \y2 k' ) dX, weakly in L (2). Theorem (5.1) now applies easily. For instance, we can take r = r' = 1, m = m' = 1, = O, p = w, ~ = 2, and 2 i=l =x..x/at V! a2x/(/ ) X( I'd cEi=l ai(tT) 7 (ax/ ci) + ((t,T) yx(tyT). i 1 2 2 Here Q c L (f), and the coefficients ai are given elements of L (F). 6. AN EXISTENCE THEOREM FOR OPTIMIZATION PROBLEMS WITH STATE EQUATIONS IN THE WEAK FORM We shall now consider the case, mentioned in 1a, where the state equations (1.2), (1.3) are written in the weak form as is customary in partial differential equations theory. We shall use the general notations of the previous sections. Let W de30

note a normed space of test functions w = (wlw2), where w1 c (Lq(G))r, w2 (L (r))r, and p + q = 1, with 1 < p < +, 1 < q < + 0, and the usual q conventions hold. We shall assume that the norms w of w in (L (G))r and lq 1 q Iw211q of w2 in (L (r)) are related to the norm lwf|W of w = (wLw2) in W by a. relation of the form Iwll q + 11w lq < KI1wl1 W (6.1) lq 2q - w where K is a constant. We shall denote by W* the dual space of W. We shall deal here with only three operators, ~, t as in 5 ~, and J replacing both and J.: S x + (L1(G))s,. S x + (L(F))S J: S x.x Q -+W*. For every x c S, u c T, ~ c, v c T, ~ c t, we consider now the operator h, or h(x,u, ~,v,, h: W + E', defined by hw = f (t,(k(x, ))(t),u(t))wl(t)dt G + f g(t,((x, ))(t),v(t))w2(t)di, where fw and gw2 denote inner products in Eand E, r espectively. Instead of state equations (.2), (.3) we shall now cnsider the unique Instead of state equations (5.2), (5-3) we shall now consider the unique state equation in the weak form = h, or (6.2) 31

~w = hw for all w e W, or (6.3) 3(xQ, @)(wl,w2) = I f(t,(N(x, ))(t),u(t))wl(t)dt G g(t,( (x, )))(t),v(t))w (t)d. r for all (wl,W2) c W. (6.4) Note that relation (6.1) implies (L p(G)) x (L())r W and, as mentioned, c5: S x ( x + W*. In most applications, however, we shall have &J: x T x T (L(G)) x (L (r)) Cw* p p and the actual determination of W* will be irrelevant. We understand here that the present unique state equation (6.3) is the weak form of state equations (5.2), (5-3). In other words, in any particular situation J and W must be so chosen so that any solution of the equations (5.2), (5.3) (strong form) is necessarily a solution of (6.3). Thus, we are interested here in the problem of the minimum of the functional I[x,u,,v, ] = f (t,(M(x,)@))(t),u(t))dt G + " go(t,((x, ~ ))(t),v(t))d~, (6.5) 32

with state equation (in the weak form) Cfw = hw for all w E W, (6.6) and constraints ~(~(x,6 ))(t) E A(t), u(t) E U(t,(lk(x, ))(t)) a.e. in G, (6.7) (M(x, ())(t) C B(t), v(t) E V(t,((xO ))(t)) i-a.e. in r. (6.8) In the present situation we shall require suitable growth conditions, or condition (H ): (H ) For p = 1 we assume that, given e > O, there are functions - > O, e L (G), and 4 > 0, 4 c Ll(r), such that I f(t,y,u)l < ~~(t) + Efo(t,yu), for all (t,y,u) C M, Ig(ty,v)l < ~ (t) + ego(t,~,v). 0 for all (t,.,v) C M. If. p > 1 we assume that there are functions 4 > 0, E 0- O o o0 L1(G), and P0 > O, EC Ll(r), and constants a > O, b > 0, such that If(t,y,u)lP < o (t) + af (t,y,u) for all (t,y,u) C M, 33

1tq,(t,3O,-1v)l" < (t) t bg (t,9Ov) for all (t,y,v) C M. This condition, for p = 1, has been systematically used by Cesari [3be] as a suitable extension of previous more restrictive growth hypotheses used by Tonelli and McShane. A triple (x,u,v), or system (x,u, ~,v, ~), is now said to be admissible provided x E S, u e T, ~ c O, v c T, ~ e ~, relations (6.6), (6.7), (6.8) hold f (t, ~(x))(t),u(t)) C L1(G), go(t,(t(x, ))(t),v(t)) c LL(F). Also, we require that f(t,(l7(x, ))(t),u(t)) c (L (G))r and that 1ep g(t,(YC(x,~ ))(t),v(t)) c (L (r))r. In the existence theorem below, however, this last requirement will be a consequence of property (H). We shall now consider nonempty closed classes 2 of admissible systems (x,u,G,v, ), where the definition of closedness is analogous to the ones in SS 4 and 5. (6.i) Existence Theorem. Let G be bounded and measurable, A, B, M, M closed, f (t,y,u), f(t,y,u) = (f,...,f ) continuous on M, go(t,,v), o 1 r o g(t,y,v) = (gl'''',gr') continuous on M, and assume that the sets Q(t,y) have property (Q) on A, and the sets R(t,j) have property (Q) on B. Let us assume that there are functions r(t) > O, t c G, E C L1(G), and t(t) > O, t C r, EC L1( ), such that f (t,y,u) > -4(t) for all (t,y,u) C M, and go(t, y,v) >' -(t) for all (t,y,v) E M. Let us assume that relation (6.1) holds, and that growth condition (H ) is satisfied. Let us assume that, for every sequence x,,, xk, (, k = 1,2,..., of elements x, xk E S 34

~' c Q, ~, < with xk - x weakly in S, ~ wakly in Q, k +- weakly in ~, it occurs that there is. some subsequence [k ] such that p X~k,uk ) -11(x,~) strongly in (L (G)) i(Xk "'k, )w+0(x, O, ~ )w for every w e W, (6.9) as p + o. Let, -be a nonempty closed class of admissible systems (x,u, ~, v, ) such that [x)Q, { }, { ~ ] ) are weakly sequentially relatively compact. Then the functional (6.1), or I[x,u, ~,v, ], has an absolute minimum in Q. The hypothesis concerning ~ above can be reworded by saying that ~(Xk ukp, v ) + +(x, ~, ~ ) in the weak star topology on W*. Proof. As usual let i be the infimum of I[x,u, ~,v, ] in the class P. Since f > -', g > - and Q is nonempty, i is finite. Let 0- 0 (xkuk(Ukvk, E VkV ), k = 1,2,.., be a sequence with I[xk, UkC VkaE] + i as k oo, and we may well assume that. = - Ixk, k vk,~ 3< = + k _< i+l, k = 1,2,... Since we have assumed that the sets (x)n, ( ~ } are weakly sequentially relatively compact, there is a subsequence, say still [k] for the sake of simplicity such that xk t x weakly in S, D + ~ weakly in 35

Q, ~ ~ weakly in Q as k r o. We may even assume that the subsequence has been so chosen that limit relations (6.9) hold. Let k(t) = f(t ( )(tuk(t)), t c G, zk(t) g(t,(xk, )(t),vk(t)), t r, k = 1,2,.... By growth condition (H ) and I[xk,uk,Evk, c] < i+l for all k, we see that, if p > 1, the functions zk(t) and zk(t), k = 1,2,..., are equibounded in the norms of (L (G)) and (L (r)), respectively. If p = 1 it follows from an argument of Cesari [3be] that the same functions zk(t) and zk(t) are equiabsolutely integrable in G and r, respectively. In any case, there exists a, subsequence, say still [k] for the sake of simplicity, and elements z(t), t c G, z C (L (G))r, and z(t), t c r, z E (L (r)) r such that p p Zk - Z weakly in (L (G)) and zk + Z weakly in (L ())r' In other words, Zk and zk weakly Lp. In other words, f(t,7(xk, ( )(t),uk(t)) + z(t) weakly in (L (G))r k)'\j~ k p g(t,(xk, c)(t),vk(t)) -+ (t) weakly in (L (r)) r (6.10) as k co, while (N(Xk~E, ))(t) E A(t), uk(t) EU(t,(1(Xk, ))(t)) a.e. in G, ((Cxk,, ))(t) c B(t), vk(t) C V(t,('(xk, S))(t)) -a.e. in r. (6.12)

If w = (wl,W2) is any element of W then, by relation (6.1) we know that 1!2 w C (L (G))r and w2 C (Lq(r)), hence f zk(t)wl (t)dt f z(t)w (t)dt, f zk(t)w2(t)dp + f Z(t)w2(t)dp. G G I as k c+. Finally, by the definition of the operator h, we have W'(t)w (t)dCi (6.13) h(Xk,k,',V k )w + f z(t)w1(t)dt + f (t)w (t)d (6.1 G 2 as k + o, for every w = (wl,w2) e W. By hypothesis we have also (~x1 in) f,W, o, @ W (6.14) as k + o, again for every w E W. Here each system (xk,uk,( k,vk, D), k = 1,2,..., is admissible, hence the first members of (6.13) and (6.14) are equal for every w C W. From (6.13) and (6.14), by comparison we obtain A(x, ~, ~ )w = f z(t)wl(t)dt + f z(t)w (t)dp (6.15) G 1 for every w C W. Now relations (6.10), (6.11), (6.12) show that we can apply lower closure theorem (3.i) with S replaced by S x O x @ and with p = r, p r'. Here the sets Q(t,y) have property (Q) on A, hence property Q(r+l) by force of (l.i). Analogously, the sets R(t,j) have property (Q) on B, hence property Q(r'+l). We conclude that (1(x,O ))(t) e A(t) a.e. in G, that M(x, ))(t) c B(t) C t-a.e. in r, and that there are elements u c T, v E T such that 37

u(t) C U(t,(A(x, ))(t)), z(t) f(t,(7l(x, ))(t)u(t)) a.e. in G, v(t) C V(t, ((x ())(t), 0(t) = g(t,( (x,( ))(t),(t t)) [i-a.e. in r, f(t,( (x, ))(t),u(t)) c L1(G) go(t,( (x, ))(t),v(t)) c Ll(r), I[x,u, ~,v, ~] < i. (6.16) Relations (6.15) and (6.16) show, by comparison, that j(x,O )w = h(x,u, v, )w for all w W. Thus, the system (x,u, ~,v, ~) is admissible, and since Q is closed, the same system belongs to Q, and I[x,u, ~,v, ~ i > i. Thus, I = i, and existence theorem (6.i) is thereby proved. Remark. Remarks 1, 2, and 3 of ~ 4 apply to the present theorem (6.i) as well. Example. Let G be an open bounded connected subset of EV, v > 1, of class K. We are concerned with the minimum of a functional I[x,u] = S fo(t,(kx)(t),u(t))dt, (6.17) G 38

with no boundary conditions, with state equations which we want to be a weak form of Lv 1 2x/(jti)2 = f(t,( c)(t),u(t)), (6.18) and with constraints (0x)(t) C A(t), u(t) c U(t,(7 )(t))j (6.19) Here x and u are functions on G. Thus, g =0, we have no boundary condition on x, we can take g = 0, J= O, y= 0, and need make no references to r, B, 0 V, M. In this problem x and u are functions on G. We shall think of W as simply being C (G) and write elements w = (w1,o) o C W as simply wl(t), t C G, with wl E C (G). As a weak form of (6.18) we take now: -f E1 (/l 1/)(wlti)dt = S f(t,(lx)(t),u)w ( t)dt (6.20) G G for all wl C C (G). It is easy to verify that any (strong) solution x, u of (6.18), say x C W2(G), u E T, is certainly a solution of (6.20). We take x E W (G), p = 1, u c T, (that is, u measurable in G). We can 00 00 take in W = C (G) any topology which is stronger than the topology on C (G) induced by the W (G) norm. For instance: -00 lw 1lw = Maxiwl(t) +ZV Max I W/tij, where each Max is taken in G. Then IlWll = i|W|lL (G) < lwilw' and thus (6.1) holds with K = 1, and W c L (G). C00 359

Let 7: S +- (L (G))S be any operator such that x, xk c S, k = 1,2,..., and xk + x weakly in S implies Pk +? strongly in (L (G)). Let A(t), A, U(t,y), M be defined as usual and closed, with r = I, m = 1, let f (t,y,u), 0 f(t,y,u) be real valued continuous functions on M, and let Q(t,y) be the sets Q(ty) = (,) E E > f(tyu), z - f(t,y,u), u C U(t,y)). We assume that the sets Q(t,y) have property (Q) in A. We shall assume that there is some function r > 0, E C L (G) such that f (t,y,u) > -t(t) for all t c G. We shall also assume that growth condition (H ) holds for f and f with p = 1. Here J: S + W* is defined by G 1 1 Now, if x, xk C S = W (G), k = 1,2,..., and xk + x weakly in S = Wi(G), k i then axk//ti ax/- t as k + a weakly in LL(G), i = 1,...,v, hence (Xk)W.i (x)wl for every w1 c L (G) and then certainly for every wl C W = Ca(G). Thus, thehypothesis required on' in (6.i) is satisfied. A pair (x,u) is here admissible provided x E W (G), u is measurable in G, (79x)(t) c A(t) and u(t) c U(t, (n)(t)) a.e. in G, f (t,(CY)(t),u(t)) E Li(G), and relation (6.20) holds for every wl E W = Ca(G). 40

Note that hypothesis (H ) (for p = 1) certainly assures that also f(t,(71)(t),u(t)) E L1(G). We shall take for S a nonempty closed class of admissible pairs (x,u). Theorem (6.i) now guarantees that the functional (6.17) with state equation (6.20) (in weak form) and constraints (6.19) has an absolute minimum in Q. 7. APPLICATIONSTO PROBLEMS OF OPTIMIZATION WITH AN EVOLUTION EQUATION IN WEAK FORM In this section we apply theorem (6.i) to problems of optimization wiznt an evolution equation in the weak form. Example 1. We are using in this and following examples the notations of the example in ~ 5. We are concerned with the problem of the minimum of the functional I[xu,~v,1~v3] = Gf (t,(,x) (t,T), u (t,)dtdT + S g(T,(%X) (T) V (T) )dT (6.21) 3 with no boundary conditions on Fl and F3, with a state equation (concerning r2 and G) which will be a suitable weak form of the system of equations ax/at Z~ a 2x/(~i)2 = f(t,T,(1x)(tT),u(tT)) in G, (6.22) 1 = 1 ~ ~~~lc

6x/jn + E (t,T)72x(t,T) = in r (6.23) and constraints (I)(t,T) c A(t,T), u(t,) E U(t,T,( 7r)(t,))) a.e. in G, (6.24) ({x)(T) c B(T), v3(T) C V(T,(4x)(T)) a.e. in r3. (6.25) 1 1 1 Here x, u: G + E,: 2, r3 + E, denote real valued functions, x state variable, u, Q v, v controls. In other words, we are interested in the determination of a function x(t,T) in G (in particular, of its initial values, say ((T) = x(o,T) on r1), and of controls u(t,T) in G, cj(t,T) in r2, v3(T) in F3, such that the functional (6.21) has its minimum value, under constraints (6.24), (6.25), and a suitable weak form of state equations (6. 22), (6.23). We take for W the space of all pairs w = (w,7w), where c c C (clG). As a weak form of (6.22-23) we take the equation fJ Z ( x/T i) (8 /ut')dtdT + f ( x/,/)oX(tT)dtdT G G + Q (t,T)72x(t,T)72c(t,T)dtdli 2 f f(t,T,(ILx)(t,T),u(t,T)w(t,T)dtdT for all co c C (G) (6.26) G Here Jw, or (x,O )w, that is, the operator ~, is defined by the first member of (6.26). It is easy to verify that any strong solution x, u, O of (6.22-23), say with x E S1, u E T, c L (r2), is certainly a solution of (6.26) for 42

all w c C (G). We shall take here x E S = Wi(G), p = 1, u c T. E c with a weakly closed subset of L (r2), which is bounded in the norm of L (r') (in 00 oo 2 other words, the elements ( E? c L (r) are equibounded in ~2). We take in W any topology which is stronger than the topology on C 0(G) x CC (r) induced by the topologies of W (G) and L (r), say 00 00 |Iw|I = 11 (W0,6)la = Max|l(t,T)I + i=l MaxI o/Ti I where all Max are taken in clG. Then, (6.1) holds with q = co, K = 2, and W c W(G) x L(r). 00.00 Let?: S - (L (G))5 and: S + (LL(r3))s' be operators such that x, xk S = w1(G), k = 1,2,..., and xk + x weakly in S implies k strongly in kk. (L1(G)~), )xk + {x strongly in (Ll(r3))s Note that xk + x weakly in S = W (G) implies that xk - x strongly in L1(G), axk/at 6 ax/3t, ~axk/ + 3x/ t weakly in L (G), i = 1,...,v, and 7xk - yx strongly in Ll(r). If,, k = 1,2,..,, are elements of w( t( 7x stron x ) lf v ( x) L (P 2) with weakly in L (r2), then V2k(Y2Xk) ~ v2(Y2x) weakly in L (r2). Finally, from the definition of c(x, ), we conclude that J(xk, 9 )w + (x, )w as k + o for every w = (L,yU) E W, as requested. For any (t,T) c G let A(t,T) c E be a given set, let A, U(t,T,y), M be defined as usual, A and M closed, let f (t,T,u), f(t,T,y,u) be real valued continuous functions on M. Let Q(t,T,y) be the sets

Q(t,T,y) = ((,z) E Ij > f(t,T,y,u), z - f(ty,y,u), u C U(t,T,y)J, and let us assume that these sets have property (Q) on A. Also, in harmony with (6.i),we assume that there is a function *(t,T) > 0, V E L (G) with f (t,T,y,u) > -*(tT) for all (t,T,y,u) c M, and that for every ~ > 0 there 0 is some function P (tT) > O, 2 c L (G), such that If(tT,yu)I < P (tT) + &f (tT,y,u) for all (t,T,y,u) C M. For any (T,T) c r3, that is, T c clG', let B(T) c E be a given set, let B, V(T~,Y), M be defined as usual, B and M closed, let go(T,,vv3) be real valued and continuous on M. We assume that there is a real valued function V3(T) > ~0, Li(G') with go(T~,'v) > -83(T) for all (T,y,v ) C M. Here g = 0. hence the corresponding sets R are half straight lines, and certainly are convex and satisfy property (Q) and the corresponding part of conditions (H ) is also satisfied. On Fl and F we have both g =, g = 0, and no ~ 2 o further discussion is needed. A system (X, YV3) is here admissible provided x c W (G), u is measurable in G, v3 measurable in F3 Q T2 C L 2 (r), f (t,T, mx)(t,T),u(tT)) cL1(G), g o(,( x)(T),v 3(T)) C LL(r), and relations (6.24), (6.25), (6.26) hold. Because of hypothesis (H ), f(t,7,(77x)(t,T),u(t,T)) also is in L (G). We shall take for 0 a nonempty closed class of admissible systems. Theorem (6.i) now guarantees that the functional (6.21) with state equation (6.26) (in weak form) and constraints (6.24), (6.25) has an absolute minimum in Q. Example 2. The same as example 1 where now we are concerned with the 44

minimum of the functional (6.21), with constraints (6.24), (6.26), state equation which will be a suitable weak form of (6.22) in G, and no boundary conditions on Fl1 F2,' F. Here x, u: G E - E E,denote real valued functions, x state variable, u, v3 controls. In other words, we are concerned with the determination of a function x(t,T) in G (in particular, of its initial values, say ( (T) = x(oT) on o1, and lateral values, (tT) = x(t,T) on F2), and of controls u(t,T) in G and v3(T) on F3, such that the functional (6.21) has its minimum value under constraints (6.24), (6.25) and suitable weak form of (6.22). 00 Here we take for W the space of all pairs w = (o,o) where w c C (G). As 0 a weak form of (6.22) we take i=l 1 (ox/'T )(ad/3T )dtdT + f ( x/tt)w(tT)dtdT G G J f(t,,( x)(t,),u(t,Y))w(t,)dtdT G for all w c C (G). The discussion of this example is similar to example 1. Examples similar to the ones above have been considered by J. L. Lions [7ab],.

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