Optimal control of nonlinear elliptic problems with sparsity

TL;DR

This paper investigates the optimal control problem for nonlinear elliptic PDEs with sparsity-promoting measures, proving existence of solutions and properties of minimizers under various conditions.

Contribution

It establishes the existence of minimizers for a sparsity-regularized control problem involving nonlinear elliptic equations and analyzes their properties.

Findings

Existence of minimizers for the control problem.

Minimizers inherit nonnegativity or boundedness when the desired state has these properties.

The control functions are measures in the space of finite Borel measures.

Abstract

We study the minimization of the cost functional \[ F(\mu) = \lVert u - u_d \rVert_{L^p(\Omega)} + \alpha \lVert \mu \rVert_{\mathcal{M}(\Omega)}, \] where the controls $\mu$ are taken in the space of finite Borel measures and $u \in W_0^{1, 1}(\Omega)$ satisfies the equation $- \Delta u + g(u) = \mu$ in the sense of distributions in $\Omega$ for a given nondecreasing continuous function $g : \mathbb{R} \to \mathbb{R}$ such that $g(0) = 0$. We prove that $F$ has a minimizer for every desired state $u_d \in L^1(\Omega)$ and every control parameter $\alpha > 0$. We then show that when $u_d$ is nonnegative or bounded, every minimizer of $F$ has the same property.