Optimal Control of Nonlinear Elliptic Problems with Sparsity

We study the minimization of the cost functional $F(\mu) = ||u-u_d||_{L^p(\Omega)}+\alpha||\mu||_{{\cal M}(\Omega)}$, where the controls $\mu$ are taken in the space of finite Borel measures and $u\inW^{1,1}_0(\Omega)$ 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