Critical Cones for Sufficient Second Order Conditions in PDE Constrained Optimization

Eduardo Casas,Mariano Mateos

doi:10.1137/19m1258244

Abstract

In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.

Highlights

Let us consider a domain \Omega \subset \BbbR n, n \leq 3, with a Lipschitz boundary \Gama
We investigate second order sufficient optimality conditions for the control problem (P) min J(u) := F (u) + \mu j(u), u\in Uad where \mu \geq 0
The main goal of this paper is to prove that a second order optimality condition based on this cone along with the first order optimality conditions imply the strong local optimality of u\=

Summary

Introduction

Let us consider a domain \Omega \subset \BbbR n, n \leq 3, with a Lipschitz boundary \Gama. Given T > 0, we denote Q = \Omega \times (0, T ) and \Sigma = \Gama \times (0, T ). We investigate second order sufficient optimality conditions for the control problem (P) min J(u) := F (u) + \mu j(u), u\in Uad where \mu \geq 0. \nu \Omega \in \{ 0, 1\} , and j : L1(Q) \rightar \BbbR is given by j(u) = \| u\| L1(Q). Yu denotes the state associated to the control u related by the following semilinear parabolic state equation:. U yu = 0 yu(0) = y0 in Q, on \Sigma , in \Omega. \ast Received by the editors April 25, 2019; accepted for publication (in revised form) December 11, 2019; published electronically February 20, 2020

EDUARDO CASAS AND MARIANO MATEOS

Assume now that u

Since u

Therefore v