Pontryagin’s Principle for Optimal Control Problems Governed by Semilinear Elliptic Equations

Abstract

In this paper we prove an extension of Pontryagin’s principle for optimal control problems governed by semilinear elliptic partial differential equations. The control takes values in a bounded subset, not necessarily convex, of some Euclidean space; the cost functional is of Lagrange type; and some general equality and inequality state constraints are imposed. To derive Pontryagin’s principle we combine a suitable penalization of the state constraints with Ekeland’s principle. In the absence of equality state constraints we establish the optimality conditions in a qualified form for “almost all” problems. In a first stage, the classical spike perturbations of the controls used to derive Pontryagin’s principle are replaced by a new type of variations.