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<datestamp>2018-01-11</datestamp>
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<metadata><dc>
<publisher>HAL CCSD</publisher>
<title lang=en>Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects</title>
<creator>Attouch, Hedy</creator>
<creator>Maingé, Paul-Emile</creator>
<contributor>Institut de Mathématiques et de Modélisation de Montpellier (I3M) ; Université Montpellier 2 - Sciences et Techniques (UM2) - Université de Montpellier (UM) - Centre National de la Recherche Scientifique (CNRS)</contributor>
<contributor>Laboratoire de Mathématiques Informatique et Applications (LAMIA) ; Université des Antilles et de la Guyane (UAG)</contributor>
<description>International audience</description>
<source>ISSN: 1292-8119</source>
<source>EISSN: 1262-3377</source>
<source>ESAIM: Control, Optimisation and Calculus of Variations</source>
<publisher>EDP Sciences</publisher>
<identifier>hal-00803774</identifier>
<identifier>https://hal.archives-ouvertes.fr/hal-00803774</identifier>
<identifier>https://hal.archives-ouvertes.fr/hal-00803774/document</identifier>
<identifier>https://hal.archives-ouvertes.fr/hal-00803774/file/ATTOUCH-MAINGE-09-04-2010-COCV.pdf</identifier>
<source>https://hal.archives-ouvertes.fr/hal-00803774</source>
<source>ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2010, 17, pp.836-857. 〈10.1051/cocv/2010024〉</source>
<identifier>DOI : 10.1051/cocv/2010024</identifier>
<relation>info:eu-repo/semantics/altIdentifier/doi/10.1051/cocv/2010024</relation>
<language>en</language>
<subject lang=en>Second-order evolution equations</subject>
<subject lang=en>asymptotic behavior</subject>
<subject lang=en>dissipative systems</subject>
<subject lang=en>maximal monotone operators</subject>
<subject lang=en>potential and non-potential operators</subject>
<subject lang=en>cocoercive operators</subject>
<subject lang=en>Tikhonov regularization</subject>
<subject lang=en>heavy ball with friction dynamical system</subject>
<subject lang=en>constrained optimization</subject>
<subject lang=en>coupled systems</subject>
<subject lang=en>dynamical games</subject>
<subject lang=en>Nash equilibria</subject>
<subject>[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]</subject>
<type>info:eu-repo/semantics/article</type>
<type>Journal articles</type>
<description lang=en>In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order damped evolution equations, which are governed by the sum of the gradient operator of a convex differentiable potential function , and a maximal monotone operator which is assumed to be cocoercive. Under a sharp condition involving the viscous and the cocoercive parameters, it is proved that each trajectory weakly converges to an equilibrium. Passing from weak to strong convergence is obtained by introducing an asymptotically vanishing Tikhonov-like regularizing term. As special cases, we recover the asymptotic analysis of the heavy ball with friction dynamic attached to a convex potential, the second-order gradient-projection dynamic, and the second-order dynamic governed by the Yosida approximation of a general maximal monotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization, dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.</description>
<date>2010-08-06</date>
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