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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. 370 (2018), 755-790
MSC (2010): Primary 34G25, 37N40, 46N10, 47H05
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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  • [1] Felipe Alvarez and Alexandre Cabot, Asymptotic selection of viscosity equilibria of semilinear evolution equations by the introduction of a slowly vanishing term, Discrete Contin. Dyn. Syst. 15 (2006), no. 3, 921-938. MR 2220756,
  • [2] H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
  • [3] Hedy Attouch, Viscosity solutions of minimization problems, SIAM J. Optim. 6 (1996), no. 3, 769-806. MR 1402205,
  • [4] Hedy Attouch, Giuseppe Buttazzo, and Gérard Michaille, Variational analysis in Sobolev and BV spaces: Applications to PDEs and optimization, 2nd ed., MOS-SIAM Series on Optimization, vol. 17, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2014. MR 3288271
  • [5] H. Attouch and R. Cominetti, A dynamical approach to convex minimization coupling approximation with the steepest descent method, J. Differential Equations 128 (1996), no. 2, 519-540. MR 1398330,
  • [6] Hedy Attouch and Marc-Olivier Czarnecki, Asymptotic behavior of coupled dynamical systems with multiscale aspects, J. Differential Equations 248 (2010), no. 6, 1315-1344. MR 2593044,
  • [7] Hédy Attouch, Marc-Olivier Czarnecki, and Juan Peypouquet, Prox-penalization and splitting methods for constrained variational problems, SIAM J. Optim. 21 (2011), no. 1, 149-173. MR 2765493,
  • [8] Hédy Attouch, Marc-Olivier Czarnecki, and Juan Peypouquet, Coupling forward-backward with penalty schemes and parallel splitting for constrained variational inequalities, SIAM J. Optim. 21 (2011), no. 4, 1251-1274. MR 2854582,
  • [9] Hédy Attouch and Alain Damlamian, Strong solutions for parabolic variational inequalities, Nonlinear Anal. 2 (1978), no. 3, 329-353. MR 512663,
  • [10] D. Azé, Eléments d'analyse convexe et variationnelle, Ellipses, Paris, 1997.
  • [11] Heinz H. Bauschke and Patrick L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. MR 2798533
  • [12] J. B. Baillon and R. Cominetti, A convergence result for nonautonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming, J. Funct. Anal. 187 (2001), no. 2, 263-273. MR 1875148,
  • [13] J. B. Baillon and H. Brezis, Une remarque sur le comportement asymptotique des semigroupes non linéaires, Houston J. Math. 2 (1976), no. 1, 5-7 (French). MR 0394328
  • [14] Heinz H. Bauschke, D. Alexander McLaren, and Hristo S. Sendov, Fitzpatrick functions: inequalities, examples, and remarks on a problem by S. Fitzpatrick, J. Convex Anal. 13 (2006), no. 3-4, 499-523. MR 2291550
  • [15] Radu Ioan Boţ and Ernö Robert Csetnek, Forward-backward and Tseng's type penalty schemes for monotone inclusion problems, Set-Valued Var. Anal. 22 (2014), no. 2, 313-331. MR 3207742,
  • [16] Radu Ioan Boţ and Ernö Robert Csetnek, Approaching the solving of constrained variational inequalities via penalty term-based dynamical systems, J. Math. Anal. Appl. 435 (2016), no. 2, 1688-1700. MR 3429667,
  • [17] H. Brézis, Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution, Lecture Notes, vol. 5, North-Holland, 1972.
  • [18] Haïm Brézis, Asymptotic behavior of some evolution systems, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977) Publ. Math. Res. Center Univ. Wisconsin, vol. 40, Academic Press, New York-London, 1978, pp. 141-154. MR 513816
  • [19] Haïm Brezis and Alain Haraux, Image d'une somme d'opérateurs monotones et applications, Israel J. Math. 23 (1976), no. 2, 165-186. MR 0399965,
  • [20] Ronald E. Bruck Jr., Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal. 18 (1975), 15-26. MR 0377609,
  • [21] Regina Sandra Burachik and B. F. Svaiter, Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10 (2002), no. 4, 297-316. MR 1934748,
  • [22] Alexandre Cabot, Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization, SIAM J. Optim. 15 (2004/05), no. 2, 555-572. MR 2144181,
  • [23] R. Cominetti, J. Peypouquet, and S. Sorin, Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization, J. Differential Equations 245 (2008), no. 12, 3753-3763. MR 2462703,
  • [24] Ivar Ekeland and Roger Témam, Convex analysis and variational problems, corrected reprint of the 1976 English edition, Classics in Applied Mathematics, vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. MR 1727362
  • [25] Simon Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 20, Austral. Nat. Univ., Canberra, 1988, pp. 59-65. MR 1009594
  • [26] Hiroyuki Furuya, Koichi Miyashiba, and Nobuyuki Kenmochi, Asymptotic behavior of solutions to a class of nonlinear evolution equations, J. Differential Equations 62 (1986), no. 1, 73-94. MR 830048,
  • [27] N. Kenmochi, Solvability of nonlinear equations with time-dependent constraints and applications, Bull. Fac. Educ. Chiba Univ. 30 (1981), 1-87.
  • [28] B. Lemaire, On the convergence of some iterative methods for convex minimization, Recent developments in optimization (Dijon, 1994) Lecture Notes in Econom. and Math. Systems, vol. 429, Springer, Berlin, 1995, pp. 252-268. MR 1358403,
  • [29] J.-E. Martínez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions, Set-Valued Anal. 13 (2005), no. 1, 21-46. MR 2128696,
  • [30] Juan-Enrique Martinez-Legaz and Michel Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2 (2001), no. 2, 243-247. MR 1848704
  • [31] Zdzisław Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. MR 0211301,
  • [32] Gregory B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl. 72 (1979), no. 2, 383-390. MR 559375,
  • [33] Jean-Paul Penot and Constantin Zălinescu, On the convergence of maximal monotone operators, Proc. Amer. Math. Soc. 134 (2006), no. 7, 1937-1946. MR 2215762,
  • [34] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
  • [35] R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362
  • [36] Behzad Djafari Rouhani, Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl. 147 (1990), no. 2, 465-476. MR 1050218,
  • [37] S. Simons and C. Zălinescu, A new proof for Rockafellar's characterization of maximal monotone operators, Proc. Amer. Math. Soc. 132 (2004), no. 10, 2969-2972. MR 2063117,
  • [38] S. Simons and C. Zălinescu, Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Anal. 6 (2005), no. 1, 1-22. MR 2138099
  • [39] Denis Torralba, Développements asymptotiques pour les méthodes d'approximation par viscosité, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 2, 123-128 (French, with English and French summaries). MR 1373747

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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

Alexandre Cabot
Affiliation: Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université Bourgogne Franche-Comté, 21000 Dijon, France

Marc-Olivier Czarnecki
Affiliation: Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France

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

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