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Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects*

Published online by Cambridge University Press:  06 August 2010

Hedy Attouch  and
Paul-Émile Maingé
Hedy Attouch
Affiliation:
Institut de Mathématiques et de Modélisation de Montpellier, UMR CNRS 5149, CC 51, Université Montpellier II, place Eugène Bataillon, 34095 Montpellier Cedex 5, France. attouch@math.univ-montp2.fr.
Paul-Émile Maingé
Affiliation:
Université des Antilles-Guyane, D.S.I., CEREGMIA, Campus de Schoelcher, 97233 Schoelcher Cedex, Martinique, France. Paul-Emile.Mainge@martinique.univ-ag.fr
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Abstract

In the setting ofa real Hilbert space ${\cal H}$, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolutionequations

          ü(t) + γ$\dot{u}$(t) + ∇ϕ(u(t)) + A(u(t)) = 0,

where ∇ϕ is the gradient operator of a convexdifferentiable potential function ϕ : ${\cal H}\to \R$, A : ${\cal H}\to {\cal H}$ is a maximal monotone operator which is assumed to beλ-cocoercive, and γ > 0 is a damping parameter.Potential and non-potential effects are associated respectively to∇ϕ and A. Under condition λγ2 > 1, it is proved that each trajectory asymptotically weaklyconverges to a zero of ∇ϕ + A. This condition, whichonly involves the non-potential operator and the dampingparameter, is sharp and consistent with time rescaling. Passingfrom weak to strong convergence of the trajectories is obtained byintroducing an asymptotically vanishing Tikhonov-like regularizingterm. As special cases, we recover the asymptotic analysis of theheavy ball with friction dynamic attached to a convex potential, thesecond-order gradient-projection dynamic, and the second-orderdynamic governed by the Yosida approximation of a general maximalmonotone 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.

Keywords

Second-order evolution equationsasymptotic behaviordissipative systemsmaximal monotone operatorspotential and non-potential operatorscocoercive operatorsTikhonov regularizationheavy ball with friction dynamical systemconstrained optimizationcoupled systemsdynamical gamesNash equilibria
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011 , pp. 836 - 857
Copyright
© EDP Sciences, SMAI, 2010

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