Parametric Sensitivity Analysis in Optimal Control of a Reaction Diffusion System. I. Solution Differentiability

Abstract

In this paper we consider a control-constrained optimal control problem governed by a system of semilinear parabolic reaction–diffusion equations. The optimal solutions are subject to perturbations of the dynamics and of the objective. We prove that local optimal solutions, as a function of the perturbation parameter, are Lipschitz continuous and directionally differentiable. We characterize the directional derivatives, also known as parametric sensitivities, as the solutions of auxiliary quadratic programming problems, i.e., linear-quadratic optimal control problems. Parametric sensitivities provide valuable information, e.g., in realtime optimal control environments.