{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T04:58:25Z","timestamp":1781153905812,"version":"3.54.1"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T00:00:00Z","timestamp":1778198400000},"content-version":"vor","delay-in-days":7,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100022107","name":"Universit\u00e9 de Limoges","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100022107","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2026,5]]},"abstract":"Abstract<\/jats:title>\n \n This article analyzes the asymptotic behavior of three classes of second-order inertial systems with vanishing damping when applied to non-potential operators. We consider the dynamics\n \n \n $$\\text {(V-AVD)}_\\alpha $$<\/jats:tex-math>\n \n \n (V-AVD)<\/mml:mtext>\n \u03b1<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ,\n \n \n $$\\text {(V-DIN-AVD)}_{\\alpha ,\\beta }$$<\/jats:tex-math>\n \n \n (V-DIN-AVD)<\/mml:mtext>\n \n \u03b1<\/mml:mi>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and\n \n \n $$\\text {(V-ISIHD)}_{\\alpha ,\\beta }$$<\/jats:tex-math>\n \n \n (V-ISIHD)<\/mml:mtext>\n \n \u03b1<\/mml:mi>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n $$V: \\mathcal {H} \\rightarrow \\mathcal {H}$$<\/jats:tex-math>\n \n \n V<\/mml:mi>\n :<\/mml:mo>\n H<\/mml:mi>\n \u2192<\/mml:mo>\n H<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a\n \n \n $$\\lambda $$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -cocoercive operator. By using a time-scale change\n \n \n $$t = \\sqrt{2(\\alpha +1)s}$$<\/jats:tex-math>\n \n \n t<\/mml:mi>\n =<\/mml:mo>\n \n \n 2<\/mml:mn>\n (<\/mml:mo>\n \u03b1<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msqrt>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , we show that as the damping parameter\n \n \n $$\\alpha \\rightarrow +\\infty $$<\/jats:tex-math>\n \n \n \u03b1<\/mml:mi>\n \u2192<\/mml:mo>\n +<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the rescaled trajectories converge uniformly on compact time intervals to solutions of first-order differential equations. Specifically, we establish convergence to the monotone flow\n \n \n $$\\dot{y}(s) + V(y(s)) = 0$$<\/jats:tex-math>\n \n \n \n y<\/mml:mi>\n \u02d9<\/mml:mo>\n <\/mml:mover>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n V<\/mml:mi>\n \n (<\/mml:mo>\n y<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for (V-AVD), to a Levenberg-Marquardt type equation\n \n \n $$\\dot{y}(s) + V(y(s)) + \\beta _0 \\frac{d}{ds}V(y(s)) = 0$$<\/jats:tex-math>\n \n \n \n y<\/mml:mi>\n \u02d9<\/mml:mo>\n <\/mml:mover>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n V<\/mml:mi>\n \n (<\/mml:mo>\n y<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n \n \u03b2<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n \n d<\/mml:mi>\n \n ds<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n V<\/mml:mi>\n \n (<\/mml:mo>\n y<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for (V-DIN-AVD) in finite dimensions under the condition\n \n \n $$\\lambda > 2\\beta _0$$<\/jats:tex-math>\n \n \n \u03bb<\/mml:mi>\n ><\/mml:mo>\n 2<\/mml:mn>\n \n \u03b2<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and to an implicit dynamics\n \n \n $$\\dot{y}(s) + V(y(s) + \\beta _0\\dot{y}(s)) = 0$$<\/jats:tex-math>\n \n \n \n y<\/mml:mi>\n \u02d9<\/mml:mo>\n <\/mml:mover>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n V<\/mml:mi>\n \n (<\/mml:mo>\n y<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n \n \u03b2<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n \n y<\/mml:mi>\n \u02d9<\/mml:mo>\n <\/mml:mover>\n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for (V-ISIHD) when\n \n \n $$x_1 = 0$$<\/jats:tex-math>\n \n \n \n x<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$2\\beta _0 < \\lambda $$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n \n \u03b2<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <<\/mml:mo>\n \u03bb<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . The proofs rely on Lyapunov-type energy estimates and provide a unified framework for understanding the high-damping limit of inertial systems in a non-potential setting. These results show that the non-potential setting cannot be treated as a mere extension of the gradient case\n \n \n $$V=\\nabla f$$<\/jats:tex-math>\n \n \n V<\/mml:mi>\n =<\/mml:mo>\n \u2207<\/mml:mi>\n f<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , but requires fundamentally new Lyapunov constructions and stability arguments beyond those available in the potential framework.\n <\/jats:p>","DOI":"10.1007\/s10957-026-02985-5","type":"journal-article","created":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T05:04:16Z","timestamp":1778216656000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Singular Perturbation Analysis of Accelerated Inertial Systems for Non-Potential Operators in the High-Damping Regime"],"prefix":"10.1007","volume":"209","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4375-0106","authenticated-orcid":false,"given":"Samir","family":"Adly","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kouegnon D.","family":"Mitchozounnou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Olivier","family":"Prot","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,8]]},"reference":[{"key":"2985_CR1","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s11228-024-00720-8","volume":"32","author":"S Adly","year":"2024","unstructured":"Adly, S., Attouch, H.: Complexity Analysis Based on Tuning the Viscosity Parameter of the Su-Boyd-Cand\u00e8s Inertial Gradient Dynamics. 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