Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, p. 836-857

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 evolution equations            ü(t) + γ$\stackrel{˙}{u}$(t) + ϕ(u(t)) + A(u(t)) = 0, where ϕ is the gradient operator of a convex differentiable potential function ϕ : $ℋ\to ℝ$, A : $ℋ\to ℋ$ is a maximal monotone operator which is assumed to be λ-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 weakly converges to a zero of ϕ + A. This condition, which only involves the non-potential operator and the damping parameter, is sharp and consistent with time rescaling. Passing from weak to strong convergence of the trajectories 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.

DOI : https://doi.org/10.1051/cocv/2010024
Classification:  34C35,  34D05,  65C25,  90C25,  90C30
Keywords: second-order evolution equations, asymptotic behavior, dissipative systems, maximal monotone operators, potential and non-potential operators, cocoercive operators, Tikhonov regularization, heavy ball with friction dynamical system, constrained optimization, coupled systems, dynamical games, Nash equilibria
@article{COCV_2011__17_3_836_0,
author = {Attouch, Hedy and Maing\'e, Paul-\'Emile},
title = {Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {17},
number = {3},
year = {2011},
pages = {836-857},
doi = {10.1051/cocv/2010024},
zbl = {1230.34051},
mrnumber = {2826982},
language = {en},
url = {http://www.numdam.org/item/COCV_2011__17_3_836_0}
}

Attouch, Hedy; Maingé, Paul-Émile. Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, pp. 836-857. doi : 10.1051/cocv/2010024. http://www.numdam.org/item/COCV_2011__17_3_836_0/

[1] S. Adly, H. Attouch and A. Cabot, Finite time stabilization of nonlinear oscillators subject to dry friction - Nonsmooth mechanics and analysis. Adv. Mech. Math. 12 (2006) 289-304. | MR 2205459

[2] F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert space. SIAM J. Control Optim. 38 (2000) 1102-1119. | MR 1760062 | Zbl 0954.34053

[3] F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14 (2004) 773-782. | MR 2085942 | Zbl 1079.90096

[4] F. Alvarez and H. Attouch, The heavy ball with friction dynamical system for convex constrained minimization problems, in Optimization, Namur (1998), Lecture Notes in Econom. Math. Systems 481, Springer, Berlin (2000) 25-35. | MR 1758015 | Zbl 0980.90062

[5] F. Alvarez and H. Attouch, An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping. Set Valued Anal. 9 (2001) 3-11. | MR 1845931 | Zbl 0991.65056

[6] F. Alvarez and H. Attouch, Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria. ESAIM: COCV 6 (2001) 539-552. | Numdam | MR 1849415 | Zbl 1004.34045

[7] F. Alvarez, H. Attouch, J. Bolte and P. Redont, A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics. J. Math. Pures Appl. 81 (2002) 747-779. | MR 1930878 | Zbl 1036.34072

[8] A.S. Antipin, Minimization of convex functions on convex sets by means of differential equations. Differ. Uravn. 30 (1994) 1475-1486 (in Russian). English translation: Diff. Equ. 30 (1994) 1365-1375. | MR 1347800 | Zbl 0852.49021

[9] H. Attouch and R. Cominetti, A dynamical approach to convex minimization coupling approximation with the steepest descent method. J. Diff. Equ. 128 (1996) 519-540. | MR 1398330 | Zbl 0886.49024

[10] H. Attouch and M.-O. Czarnecki, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Diff. Equ. 179 (2002) 278-310. | MR 1883745 | Zbl 1007.34049

[11] H. Attouch and A. Soubeyran, Inertia and reactivity in decision making as cognitive variational inequalities. J. Convex. Anal. 13 (2006) 207-224. | MR 2252229 | Zbl 1138.91370

[12] H. Attouch, D. Aze and R. Wets, Convergence of convex-concave saddle functions: Applications to convex programming and mechanics. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5 (1988) 537-572. | Numdam | MR 978671 | Zbl 0667.49009

[13] H. Attouch, X. Goudou and P. Redont, The heavy ball with friction method: The continuous dynamical system. Global exploration of local minima by asymptotic analysis of a dissipative dynamical system. Commun. Contemp. Math. 1 (2000) 1-34. | MR 1753136 | Zbl 0983.37016

[14] H. Attouch, A. Cabot and P. Redont, The dynamics of elastic shocks via epigraphical regularization of a differential inclusion. Barrier and penalty approximations. Adv. Math. Sci. Appl. 12 (2002) 273-306. | MR 1909449 | Zbl 1038.49029

[15] H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PDE's. J. Convex Anal. 15 (2008) 485-506. | MR 2431407 | Zbl 1154.65044

[16] J.-B. Baillon and G. Haddad, Quelques propriétés des opérateurs angles-bornés et n-cycliquement monotones. Israel J. Math. 26 (1977) 137-150. | MR 500279 | Zbl 0352.47023

[17] J.-B. Baillon and A. Haraux, Comportement à l'infini pour les équations d'évolution avec forcing périodique. Arch. Rat. Mech. Anal. 67 (1977) 101-109. | MR 493553 | Zbl 0382.47021

[18] J. Bolte, Continuous gradient projection method in Hilbert spaces. J. Optim. Theory Appl. 119 (2003) 235-259. | MR 2028993 | Zbl 1055.90069

[19] J. Bolte and M. Teboulle, Barrier operators and associated gradient-like dynamical systems for constrained minimization problems. SIAM J. Control Optim. 42 (2003) 1266-1292. | MR 2044795 | Zbl 1051.49010

[20] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Mathematical Studies. North-Holland (1973). | MR 348562 | Zbl 0252.47055

[21] A. Cabot, Inertial gradient-like dynamical system controlled by a stabilizing term. J. Optim. Theory Appl. 120 (2004) 275-303. | MR 2044898 | Zbl 1070.90107

[22] T. Cazenave and A. Haraux, An introduction to semilinear evolution equations, Oxford Lecture Series in Mathematics and its Applications 13. Oxford University Press, Oxford (1998). | MR 1691574 | Zbl 0926.35049

[23] P.L. Combettes and S.A. Hirstoaga, Visco-penalization of the sum of two operators. Nonlinear Anal. 69 (2008) 579-591. | MR 2426274 | Zbl 1168.47040

[24] R. Cominetti, J. Peypouquet and S. Sorin, Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization. J. Diff. Equ. 245 (2008) 3753-3763. | MR 2462703 | Zbl 1169.34045

[25] S. Ervedoza and E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91 (2009) 20-48. | MR 2487899 | Zbl 1163.74019

[26] S.D. Flam and J. Morgan, Newtonian mechanics and Nash play. Int. Game Theory Rev. 6 (2004) 181-194. | MR 2071364 | Zbl 1087.91001

[27] I. Gallagher, Asymptotics of the solutions of hyperbolic equations with a skew-symmetric perturbation. J. Diff. Equ. 150 (1998) 363-384. | MR 1658597 | Zbl 0921.35095

[28] J.K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE. Z. Angew. Math. Phys. 43 (1992) 63-125. | MR 1149371 | Zbl 0751.58033

[29] A. Haraux, Systèmes dynamiques dissipatifs et applications 17. Masson, RMA (1991). | MR 1084372 | Zbl 0726.58001

[30] J. Hofbauer and S. Sorin, Best response dynamics for continuous zero-sum games. Discrete Continuous Dyn. Syst. Ser. B 6 (2006) 215-224. | MR 2172204 | Zbl 1183.91006

[31] P.E. Maingé, Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 344 (2008) 876-887. | MR 2426316 | Zbl 1146.47042

[32] D. Monderer and L.S. Shapley, Potential Games. Games Econ. Behav. 14 (1996) 124-143. | MR 1393599 | Zbl 0862.90137

[33] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967) 591-597. | MR 211301 | Zbl 0179.19902

[34] B.T. Polyak, Introduction to Optimization. Optimization Software, New York (1987). | MR 1099605 | Zbl 0652.49002

[35] R.T. Rockafellar, Monotone operators associated with saddle-functions and mini-max problems, in Nonlinear operators and nonlinear equations of evolution in Banach spaces 2, 18th Proceedings of Symposia in Pure Mathematics, F.E. Browder Ed., American Mathematical Society (1976) 241-250. | MR 285942 | Zbl 0237.47030

[36] M. Schatzman, A class of nonlinear differential equations of second order in time. Nonlinear Anal. 2 (1978) 355-373. | MR 512664 | Zbl 0382.34003