{"paper":{"title":"Some properties of the value function and its level sets for affine control systems with quadratic cost","license":"","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Emmanuel Tr\\'elat (IMB)","submitted_at":"2006-07-18T15:20:48Z","abstract_excerpt":"Let $T>0$ fixed. We consider the optimal control problem for analytic affine systems: $\\ds{\\dot{x}=f\\_0(x)+\\sum\\_{i=1}^m u\\_if\\_i(x)}$, with a cost of the form: $\\ds{C(u)=\\int\\_0^T \\sum\\_{i=1}^m u\\_i^2(t)dt}$. For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function $S$ is subanalytic. Secondly we prove that if there exists an abnormal minimizer of corank 1 then the set of end-points of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607424","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"}