  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.
