Optimal control of nonlinear elliptic problems with sparsity

Abstract

We study the minimization of the cost functional \[ F(μ) = \lVert u - u_d \rVert_{L^p(Ω)} + α\lVert μ\rVert_{\mathcal{M}(Ω)}, \] 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.

25 pages

References in corpus (1)

• Schroedinger operators involving singular potentials and measure data