Second Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems

J. Frederic Bonnans (1)

1 PROMATH - Mathematical Programming

This paper presents a second-order analysis for a simple model optimal control problem of a partial differential equation, namely a well-posed semilinear elliptic system with constraints on the control variable only. The cost to be minimized is a standard quadratic functional. Assuming the feasible set to be polyhedric, we state necessary and sufficient second order optimality conditions, including a characterization of the quadratic growth condition. Assuming that the second order sufficient condition holds, we give a formula for the second order expansion of the value of the problem as well as the directional derivative of the optimal control, when the cost function is perturbed. Then we partially extend these results to the case of vector valued controls when the feasible set is defined by local and smooth convex constraints. When the space dimension $n$ is greater than 3, the results are based on a two norms approach, involving spaces $L^2(Øm)$ and $L^s(Øm)$, with $s>n/2$.

Mots clés: OPTIMAL CONTROL; ELLIPTIC SYSTEMS; SENSITIVITY ANALYSIS; EXPANSION OF SOLUTIONS; SECOND ORDER OPTIMALITY CONDITIONS; LEGENDRE FORMS; POLYHEDRICITY; TWO NORMS APPROACH

Domaines: Autre [cs.OH]; Optimisation et contrôle [math.OC]

J. Frederic Bonnans. Second Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems. [Research Report] RR-3014, INRIA. 1996. ⟨inria-00073680⟩