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Ponce, Nicolas Wilmet","submitted_at":"2017-12-17T19:17:02Z","abstract_excerpt":"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 $\\alpha > 0$. 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