We study the minimization of the cost functional F (mu) = parallel to u - u(d)parallel to(Lp(Omega)) + alpha parallel to mu parallel to(M(Omega)), where the controls mu are taken in the space of finite Borel measures and u is an element of 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 : R -> R such that g (0) = 0. We prove that F has a minimizer for every desired state u(d) is an element of 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.