{"paper":{"title":"$\\mathcal{L}^1$ limit solutions for control systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.CA","authors_text":"Franco Rampazzo, M. Soledad Aronna","submitted_at":"2014-01-01T20:50:56Z","abstract_excerpt":"For a control Cauchy problem $$\\dot x= {f}(t,x,u,v) +\\sum_{\\alpha=1}^m g_\\alpha(x) \\dot u_\\alpha,\\quad x(a)=\\bar x, $$ on an interval $[a,b]$, we propose a notion of limit solution $x,$ verifying the following properties: i) $x$ is defined for $\\mathcal{L}^1$ (impulsive) inputs $u$ and for standard, bounded measurable, controls $v$; ii) in the commutative case (i.e. when $[g_{\\alpha},g_{\\beta}]\\equiv 0,$ for all $\\alpha,\\beta=1,...,m$), $x$ coincides with the solution one can obtain via the change of coordinates that makes the $g_\\alpha$ simultaneously constant; iii) $x$ subsumes former concep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0328","kind":"arxiv","version":2},"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"}