PARETO OPTIMALITY FOR NONLINEAR INFINITE DIMENSIONAL CONTROL SYSTEMS

In this note we establish the existence of Pareto optimal solutions for nonlinear, infinite dimensional control systems with state dependent control constraints and an integral criterion taking values in a separable, reflexive Banach lattice. An example is also presented in detail. Our result extends earlier ones obtained by Cesari and Suryanarayana.


INTRODUCTION
In recent years there has been an increasing interest in optimization problems with multiple objectives conflicting with one another.The subject has its origins in mathematical economics and in particular in welfare theory and from there it passed into other subjects like game theory, operations research, optimization and optimal control.Such problems, in the context of optimal control theory, were recently considered by Cesari and Suryanarayana in a series of interesting papers [5], [6], [7].We should also mention the earlier work of Olech [9], who, motivated from the fundamental work of Cesari [4], studied similar problems in IRn.
The aim of this note is to extend the finite dimensional existence result for Pareto solutions of Cesari-Suryanarayana [5], to infinite dimensional control systems.Exploiting some recent results on the extended Fatou's lemma, obtained by Balder [1] and Papageorgiou [11], we are able to prove the closedness of a certain orientor field and through that establish the existence of Pareto optimal solutions.
A multifunction (set valued function) F: -, Pf(X) is said to be measurable if and only if for all y X, w-d(y, F(w)) inf{lly-xll: x F(w)} is measurable.If there exists a a-finite measure (.), with respect to which E is complete, then the above definition of measurability is equivalent to saying that GrF {(w,x) e X: xeF(w)} ZB(X), with B(X) being the Borel a-field of X (graph measurability).For further details on measurable multifunctions we refer to the survey paper of Wagner [16].By S we will denote the set of selectors of F(.), that belong in the Lebesgue-Bochner space LI(x) i.e. S {f LI(x) f(w) F(w) #-a.e.}.This set may be empty.However if F(. is integrably bounded (i.e.F(.) is measurable and w IF(w)l sup{llxll" x F(w)} e L), then S # q).Using S we can defineaset valuedintegralfor F(.) bysetting F(w)d(w)= {I f(w)d(w)" f S}.
Finally, in the next section we will be using some notions and results from the theory of ordered vector spaces.For the necessary background we refer to the books of Peressini [14] and Schaefer [15].
3. EXISTENCE THEOREM Let Y be a locally convex vector space with a partial order induced by a nonempty, closed, convex and pointed cone Y+.For y, y' e Y' we write y _< y' if and only if y'-y Y+.Let A c_ Y.A vector x 0 K is said to be Pareto efficient for A, if (x0-Y+) N ), where Y+ Y+\(0}.So the Pareto efficient (or Pareto optimal) points of A, are those points of which are minimal for the partial order induced by Y/.The set of Pareto efficient points of A will be denoted by Eft(A).
The class of Daniell ordered spaces includes the following ones: (a) All ordered vector spaces which have compact order intervals (resp.weakly compact, if the order is normal).
(b) All semi-reflexive ordered vector spaces, with normal order.
(c) All ordered vector spaces, with Y/ complete and having a bounded base B s.t. 0I t (in particular then, if Y/ is locally compact).
(d) All (countably) order complete Banach lattices, unless they contain a lattice isomorphic to .
Note that every (countably) Daniell space is (countably) order complete.
The following existence result is well known among people working in Pareto optimization (see for example Penot [13]).
PROPOSITION" If (Y,Y/) is a Daniell vector space and A c_ Y is nonempty and bounded below, then Eft(A) # ).Now let T [0,b], a bounded, closed interval in JR+, X a separable Banach space (the state space), Z another separable Banach space (the control space) and Y a separable, reflexive, order complete, Banach lattice.
We will consider the following infinite dimensional, nonlinear control system: By a solution of this system, we will understand a mild (integral) solution.A pair of functions x(.) e C(T,X) and u(.) LI(z), that satisfy the dynamic constraints (*), are said to be an "admissible pair".In particular x(.) is an "admissible trajectory", while u(.) is an "admissible control".We will denote the set of admissible pairs by A(x0).Finally note that system (*) has feedback type constraints, since the multifunction U(.,-) depends also on the state.
To this control system, we associate a Y-valued cost criterion of the following form: Our goal is to prove a theorem saying that every vector in Eff(J(A(x0))) is realized by an admissible pair.
To this end, we need the following set of hypotheses on the data of the problem.
H(L): L: TxXxZ Y is a measurable function.H(Q): For every (t,x) e TxX, the set Q(t,x) {(v,r/) e XY: v f(t,x,u), u U(t,x), L(t,x,u) < r/} is convex and x Q(t,x) is u.s.c.i, from X into X w.
Hb: J(A(x0) is order bounded in X.
Hypothesis H a is a controllability type hypothesis, while hypothesis H b is satisfied if for example IL(t,x,u)l < a'(t) + b'(t) (llxll + Ilull) a.e. with a'(.), b'(.)LI(y+).Recall lYl =Y++Y-- , H a and H b hold, then Eff(J(A(x0))) # ) and every element in this set can be realized by an admissible "state-control" pair.PROOF: Recalling that a reflexive Banach lattice is a Daniell space and using hypothesis H b and the proposition, we deduce that Eff(J(h(x0))) # ).
Next observe that for all k>l and all T xk(t S(t,0) x 0 / J S(t,s) V(s) ds 0 where V(s) {xe X" [[x[[ _< a(s) + b(s) (M + [W[)}.So V(s) is almost everywhere bounded, closed and since S(t,s) is compact for t-s > 0, S(t,s) V(s) is almost everywhere compact and clearly measurable in s.So by RadstrSm's embedding theorem (see for example [11]), we get that I S(t,s) V(s) ds Pkc(X), == {xk(t)}k> Pkc(X).So invoking the 0 Arzela-Ascoli theorem, we deduce that {Xk(.)}k>l is relatively compact in C(T,X).
Observe that {Uk}k> c__ S and the latter is w-compact in LI(z) (see proposition 3.1 of [10]), and by the Eberlein-Smulian theorem is sequentially w-compact.So by passing to a subsequence if necessary, we may assume that x kx in C(T,X) and W and f(t,.,.)is continuous on XxZw into Xw, we deduce that K(t)e Pwkc(X) a.e. and clearly is measurable.So proposition 3.1 of [10] tells us that SI(.) is sequentially w-compact in LI(x).Hence we may assume that v k v in LI(x) and furthermore from theorem 3.1 of [12], we have v(t) conv w-ff {vk(t)}k>_l a.e.. Thus we have (v(t), y(t)) e conv w-l] (vk(t), /k(t)) a.e.
Next let A A with D(A) H(V) n H2(V).It is well known (see for example Barbu [3]) that A generates a semigroup of contractions S(t)" L2(V) L2(V) which are compact for > 0. So we have satisfied hy pothesis H(A).

H a
This assumption implies that H a is satisfied.Finally note that because of H(L)'(3), H b is satisfied too.Rewrite (**) in the following abstract form: jb L(t,x(t), u(t)) dt-inf f s.t.i(t) Ax(t) + (t,x(t))u (t)   x(0) x 0 u(t) W a.e.This is a special case of the optimal control problem studied in section 3.So we can invoke theorem and get the following existence result.