{"paper":{"title":"Asymptotic for the perturbed heavy ball system with vanishing damping term","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Mounir Balti, Ramzi May","submitted_at":"2016-09-01T07:42:14Z","abstract_excerpt":"We investigate the long time behavior of solutions to the differential equation $\\ddot{x}(t)+\\frac{c}{\\left( t+1\\right) ^{\\alpha}}\\dot{x}(t)+\\nabla \\Phi\\left( x(t)\\right) =g(t),~t\\geq0, $ where $c$ is nonnegative constant, $\\alpha\\in\\lbrack0,1[,$ $\\Phi$ is a $C^{1}$ convex function on a Hilbert space $\\mathcal{H}$ and $g\\in L^{1} (0,+\\infty;\\mathcal{H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ as $t\\rightarrow+\\infty$ to a minimizer of the function $\\Phi$ if one exists."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00135","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"}