Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

Eduardo Casas,Fredi Tröltzsch

doi:10.1007/s10013-015-0175-6

Abstract

Second-order sufficient optimality conditions are considered for a simplified class of semilinear parabolic equations with quadratic objective functional including distributed and terminal observation. Main emphasis is laid on problems where the objective functional does not include a Tikhonov regularization term. Here, standard second-order conditions cannot be expected to hold. For this case, new second-order conditions are established that are based on different types of critical cones. Depending on the choice of this cones, the second-order conditions are sufficient for local minima that are weak or strong in the sense of calculus of variations.

Highlights

In this paper, we survey second-order optimality conditions for the following slightly simplified class of optimal control problems:M inimize (y(x, t) − yQ(x, t))2 dxdt + Q γ 2(y(x, T ) − yT (x))2 dx Ω+ ν u2(x, t) dxdt 2Q subject to the parabolic state equation ∂y (x, ∂t t) − ∆y(x, t) + R(x, t, y(x, t))
We became interested in this degenerate case by numerical observations: The numerical solution method of optimal control problems for the more general FitzHugh-Nagumo system, a well known system of mathematical physics that includes a second linear partial differential equation, turned out to be surprisingly stable for very small regularization parameters ν > 0
From the above relations we deduce that the optimal control uν is bang-bang if the set of points of Q where φν vanishes has a zero Lebesgue measure

Summary

Introduction

We survey second-order optimality conditions for the following slightly simplified class of optimal control problems:. We became interested in this degenerate case by numerical observations: The numerical solution method of optimal control problems for the more general FitzHugh-Nagumo system, a well known system of mathematical physics that includes a second linear partial differential equation, turned out to be surprisingly stable for very small regularization parameters ν > 0. This observation brought us to the ques tion, whether we can prove stability of locally optimal solutions as ν 0. We present our theory of second-order conditions here for a simpler equation but with more general nonlinearity R and for all dimensions N ∈ N

Well posedness of state equation and control problem

First-Order Necessary Optimality Conditions

Second Order Optimality Conditions

Applications to the Stability Analysis with Respect to Perturbations

Perturbation of yQ and yT

Tikhonov parameter tending to zero