{"paper":{"title":"Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Hedy Attouch, Marc-Olivier Czarnecki","submitted_at":"2016-01-31T11:47:01Z","abstract_excerpt":"In a Hilbert space $\\mathcal H$, we study the asymptotic behaviour, as time variable $t$ goes to $+\\infty$, of nonautonomous gradient-like dynamical systems involving inertia and multiscale features.\n  Given $\\mathcal H$ a general Hilbert space, $\\Phi: \\mathcal H \\rightarrow \\mathbb R$ and $\\Psi: \\mathcal H \\rightarrow \\mathbb R$ two convex differentiable functions, $\\gamma$ a positive damping parameter, and $\\epsilon (t)$ a function of $t$ which tends to zero as $t$ goes to $+\\infty$, we consider the second-order differential equation $$\\ddot{x}(t) + \\gamma \\dot{x}(t) + \\nabla \\Phi (x(t)) + \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00232","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"}