Pontryagin Maximum Principle for Optimal Control of

In this paper we investigate optimal control problems governed by variational inequalities. We present a method for deriving optimality conditions in the form of Pontryagin's principle. The main tools used are the Ekeland's variational principle combined with penalization and spike variation techniques. 1. Introduction. The purpose of this paper is to present a method for deriving a Pontryagin type maximum principle as a rst order necessary condition of optimal controls for problems governed by variational inequalities. We allow various kinds of constraints to be imposed on the state. To be more precise, we consider the following variational inequality @y @t + Ay + f(y) + @'(y) 3 u in Q = ]0; T ; (1.1a) y = 0 on = ?]0; T ; (1.1b) y(0) = y o in : (1.1c) where R n , T > 0, u is a distributed control, A is a second order elliptic operator and @y @t denotes the derivative of y with respect to t; @'(y) is the subdiierential of the function ' at y. We shall give all the deenitions we need in Section 3 and (1.1) will be made precise as well. The control variable u and the state variable y must satisfy constraints of the form u 2 U ad = f u 2 L p (Q) j u(x; t) 2 K U (x; t) a.e. in Q g L p (Q); (1.2a) where K U is a measurable set-valued mapping from Q with closed values in P(R) (P(R) being the set of all subsets of R), (y) 2 C (1.2b) with 1 < p < 1, is a C 1 mapping from C(Q) into C(Q), C C(Q) is a closed convex subset with nite codimension. The control problem is (P)