Maximum principle for optimal control of non-well posed elliptic differential equations

In this paper, we shall study two optimal control problems (P1) and (P2) governed by some semilinear elliptic di%erential equations which can admit more than one solution. We shall call such systems non-well posed systems. We obtain the Pontryagin maximum principle for problems (P1) and (P2). In the 6rst problem (P1), the cost functional may be non-smooth and the set of controls is convex while in the second problem (P2), the cost functional is smooth and the set of controls is non-convex. In both problem (P1) and problem (P2), we consider some state constraint of integral type. Due to some practical interests, many authors studied optimal control for elliptic di%erential equations. Here, we cite [1–10,12–15]. However, most of these works, except for [7,10,13], deal with state equations which are governed by monotone operators and admit a unique solution corresponding to each control. In such problems, the Pontryagin maximum principle was obtained by studying the variations of the state with respect to some perturbations of the control (cf. [1,2,12,15]). In the present work, the state equations are governed by non-monotone operators and can admit more than one solution. We cannot obtain the variations of the state with respect to the control with the same methods in [1,2,12,15]. When the state equations admit more than one solution, we even do not know how to make the sensitivity analysis of the state with respect to the control.