{"paper":{"title":"Optimal control of nonlinear elliptic problems with sparsity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.AP","authors_text":"Augusto C. 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 $\\alpha > 0$. We then show that when $u_d$ is n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.06159","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}