{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T06:52:40Z","timestamp":1787381560833,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2002,1]]},"abstract":"<jats:p>Let $\\phi : H\\rightarrow \\R$ be a $\\mathcal{C}^1$ function on a real Hilbert space H and let $\\gamma &gt; 0$ be a positive damping parameter. For any repulsive potential $V:H\\to \\R_+$ and any control function $\\varepsilon :\\R_+\\to \\R_+$ which tends to zero as $t\\to +\\infty$, we study the asymptotic behavior of the trajectories of the coupled dissipative system of nonlinear oscillators $$ \\left \\{ \\begin{array}{l} \\ddot{x}+\\gamma \\dot{x}+\\nabla\\phi(x)+\\varepsilon(t)\\nabla V(x-y)=0,\\\\ \\ddot{y}+\\gamma \\dot{y}+\\nabla\\phi(y)-\\varepsilon(t)\\nabla V(x-y)=0. \\end{array} \\right. \\leqno{\\rm (HBFC^2)} $$ We first provide general existence results and show that $\\nabla \\phi (x(t))\\to 0$ and $\\nabla \\phi (y(t))\\to 0$ when $t\\to +\\infty$, assuming either that the trajectory (x,y) is bounded, or that the potential V is bounded and that $\\phi$ satisfies the following limit condition: \\vspace*{\\abovedisplayskip}<\/jats:p>\n                  <jats:p>\\begin{itemize} \\item[\\rm (LIM)] For every sequence $ (z_n)\\subset H $ such that $ \\lim_{n\\to +\\infty}|z_n|=+\\infty,$ there exists a subsequence $(z_{\\varphi(n)})$ such that $$ \\lim_{n \\to +\\infty}{\\phi}(z_{\\varphi(n)})=+\\infty \\qquad \\mbox{or} \\qquad \\lim_{n \\to +\\infty}\\nabla{\\phi}(z_{\\varphi(n)})=0. $$ \\end{itemize}<\/jats:p>\n                  <jats:p>\\noindent If $\\varepsilon(t)$ does not tend to zero too rapidly as $t\\to +\\infty$, then the term $\\varepsilon(t) \\nabla V(x-y)$ asymptotically repulses the trajectories one from the other. Precisely, when $H=\\R$, and if $\\varepsilon$ is a ``slow' control, i.e., $\\int_0^{+\\infty} \\varepsilon(t)dt=+\\infty$, then the trajectories x and y converge to extremal points of the set $S=\\{\\lambda\\in \\R, \\nabla\\phi(\\lambda)=0\\}$ of the equilibria of $\\phi$ (when $S\\ne \\emptyset$), or they have the same limit. In particular, when S is reduced to an interval---for example, if $\\phi$ is convex---this allows us to obtain a global description of the set S. We provide numerical experiments which make our convergence results more precise.<\/jats:p>","DOI":"10.1137\/s0363012901385198","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1254-1280","source":"Crossref","is-referenced-by-count":7,"title":["Asymptotic Control of Pairs of Oscillators Coupled by a Repulsion, with Nonisolated Equilibria I: The Regular Case"],"prefix":"10.1137","volume":"41","author":[{"given":"Alexandre","family":"Cabot","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marc-Olivier","family":"Czarnecki","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,26]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998335802"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1011253113155"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1996.0104"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.2001.4034"},{"key":"R5","first-page":"273","volume":"12","author":"Attouch H.","year":"2002","journal-title":"Adv. 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Lojasiewicz,\n                      Ensembles semi\u2010analytiques\n                      , notes, Bures\u2010sur Yvette, Institut des Hautes Etudes Scientifiques, 1965."},{"key":"R21","volume-title":"Applied shape optimization for fluids","author":"Mohammadi B.","year":"2001"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1967-11761-0"},{"key":"R23","unstructured":"P. Redont,\n                      Equation de la boule pesante avec frottement: Exemple de solution non convergente\n                      , Pr\u00e9publication 99, D\u00e9partement de Math\u00e9matiques, Universit\u00e9 de Montpellier II; available online from http:\/\/www.math.univ\u2010montp2.fr."},{"key":"R24","unstructured":"A. N. Tikhonov and V. Ya. 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