In this paper we study the optimal control of systems driven by nonlinear elliptic partial differential equations. First, with the aid of an appropriate convexity hypothesis we establish the existence of optimal admissible pairs. Then we drop the convexity hypothesis and we pass to the larger relaxed system. First we consider a relaxed system based on the Gamkrelidze-Warga approach, in which the controls are transition probabilities. We show that this relaxed problem has always had a solution and the value of the problem is that of the original one. We also introduce two alternative formulations of the relaxed problem (one of them control free), which we show that they are both equivalent to the first one. Then we compare those relaxed problems, with that of Buttazzo which is based on the Λ-regularization of the “extended” cost functional. Finally, using a powerful multiplier rule of Ioffe-Tichomirov, we derive necessary conditions for optimality in systems with inequality state constraints.
Differential Equation Partial Differential Equation Alternative Formulation State Constraint Elliptic System
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