Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls

Eduardo Casas,Fredi Tröltzsch

doi:10.1137/130917314

Abstract

An optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state. The main novelty of the paper is the presence of the $L^1$-norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions. Second-order sufficient optimality conditions are analyzed. They are applied to show the convergence of optimal solutions for vanishing $L^2$-regularization parameter for the control. The associated convergence rate is estimated.

Highlights

We study the optimal control problem (Pν )
Uad = {u ∈ L∞(Ω) : α ≤ u(x) ≤ β for a.a. x ∈ Ω, |yu(x)| ≤ γ ∀x ∈ Ω}, and yu is the solution of the Dirichlet problem
Thanks to the presence of the L1-norm in the objective functional, a convex but not differentiable functional is to be minimized. This term accounts for sparsity of optimal controls: With increasing parameter κ, the support of the optimal controls shrinks to have the measure zero for sufficiently large κ

Summary

Introduction

|u(x)| dx, Uad = {u ∈ L∞(Ω) : α ≤ u(x) ≤ β for a.a. x ∈ Ω , |yu(x)| ≤ γ ∀x ∈ Ω}, and yu is the solution of the Dirichlet problem (1.2). The Dirichlet problem is considered in a bounded Lipschitz domain Ω ⊂ Rn, n ∈ {2, 3}, with boundary Γ = ∂Ω. Thanks to the presence of the L1-norm in the objective functional, a convex but not differentiable functional is to be minimized. This term accounts for sparsity of optimal controls: With increasing parameter κ, the support of the optimal controls shrinks to have the measure zero for sufficiently large κ

STATE CONSTRAINED SPARSE CONTROLS

Let us define

Cuν instead