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Asymptotic behavior of nonautonomous monotone and subgradient evolution equations


Authors: Hedy Attouch, Alexandre Cabot and Marc-Olivier Czarnecki
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 34G25, 37N40, 46N10, 47H05
DOI: https://doi.org/10.1090/tran/6965
Published electronically: July 13, 2017
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Abstract: In a Hilbert setting $ H$, we study the asymptotic behavior of the trajectories of nonautonomous evolution equations $ \dot x(t)+A_t(x(t))\ni 0$, where for each $ t\geq 0$, $ A_t:H\rightrightarrows H$ denotes a maximal monotone operator. We provide general conditions guaranteeing the weak ergodic convergence of each trajectory $ x(\cdot )$ to a zero of a limit maximal monotone operator $ A_\infty $ as the time variable $ t$ tends to $ +\infty $. The crucial point is to use the Brézis-Haraux function, or equivalently the Fitzpatrick function, to express at which rate the excess of $ \mathrm {gph} A_\infty $ over $ \mathrm {gph} A_t$ tends to zero. This approach gives a sharp and unifying view of this subject. In the case of operators $ A_t= \partial \phi _t$ which are subdifferentials of proper closed convex functions $ \phi _t$, we show convergence results for the trajectories. Then, we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions. Illustrations are given in the field of coupled systems and partial differential equations.


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Additional Information

Hedy Attouch
Affiliation: Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France
Email: hedy.attouch@univ-montp2.fr

Alexandre Cabot
Affiliation: Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université Bourgogne Franche-Comté, 21000 Dijon, France
Email: alexandre.cabot@u-bourgogne.fr

Marc-Olivier Czarnecki
Affiliation: Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France
Email: marco@math.univ-montp2.fr

DOI: https://doi.org/10.1090/tran/6965
Keywords: Nonautonomous monotone inclusion, subgradient inclusion, multiscale gradient system, hierarchical minimization, asymptotic behavior, Br\'ezis-Haraux function, Fitzpatrick function
Received by editor(s): July 30, 2015
Received by editor(s) in revised form: April 18, 2016
Published electronically: July 13, 2017
Additional Notes: The first author’s effort was sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant No. FA9550-14-1-0056
The authors were supported by ECOS grant No. C13E03
Article copyright: © Copyright 2017 American Mathematical Society