Second order optimality conditions for optimal control of quasilinear parabolic equations

Lucas Bonifacius,Ira Neitzel

doi:10.3934/mcrf.2018001

Abstract

We discuss an optimal control problem governed by a quasilinear parabolic PDE including mixed boundary conditions and Neumann boundary control, as well as distributed control. Second order necessary and sufficient optimality conditions are derived. The latter leads to a quadratic growth condition without two-norm discrepancy.Furthermore, maximal parabolic regularity of the state equation in Bessel-potential spaces $H_D^{-\zeta,p}$ with uniform bound on the norm of the solution operator is proved and used to derive stability results with respect to perturbations of the nonlinear differential operator.

Highlights

We discuss an optimal control problem governed by a quasilinear parabolic partial di↵erential equations (PDEs) including mixed boundary conditions and Neumann boundary control, as well as distributed control
Maximal parabolic regularity of the state equation in Bessel-potential spaces HD⇣,p with uniform bound on the norm of the solution operator is proved and used to derive stability results with respect to perturbations of the nonlinear di↵erential operator
This article is concerned with optimal control problems governed by quasilinear parabolic partial di↵erential equations (PDEs)

Summary

Introduction

This article is concerned with optimal control problems governed by quasilinear parabolic partial di↵erential equations (PDEs). First order necessary conditions for a quasilinear equation subject to integral state constraints have been proved in [30] with distributed controls in L2((0, T ) ⇥ ⌦). It is worth mentioning, that in the latter study all the coe cients of the elliptic operator may depend on u, ru and the control q. Adapting the ideas of Casas and Troltzsch from [15, 17], we prove second order necessary as well as su cient optimality conditions for Neumann boundary control in spatial dimension two and purely time-dependent control and distributed control in dimensions two and three. Some interesting but technical results are collected in the appendix

Notation and assumptions

The quasilinear parabolic state equation

Optimal control problem

Application to stability analysis