Optimal control of semilinear elliptic equation with state constraint : maximum principle for minimizing sequence, regularity, normality, sensitivity

This article deals with state constrained optimal control problem for semilinear elliptic equation in a domain Omega. The state constraint is lumped on the compactum X contained in/implied by Omega n and contains a functional parameter q in C(X ). It is shown that any minimizing approximate solution (m.a.s.) in the sense of J. Warga satisfies the pointwise maximum principle (the maximum principle for m.a.s.) if the problem is meaningful, i.e., the value of the problem is finite. It is also shown that a condition of Slater's type is sufficient for the normality in the so-called "linear-convex" problem, and the normality of the problem for some fixed value of the parameter q in C(X ) implies the Lipschitz continuity of its value function in a neighborhood of q. The paper contains illustrative examples.