Summary
We investigate the behavior at infinity of a special dissipative system, which consists of two steepest descent equations coupled by a non-autonomous conservative repulsion. The repulsion term is parametrized in time by an asymptotically vanishing factor. We show that under a simple slow parametrization assumption the limit points, if any, must satisfy an optimality condition involving the repulsion potential. Under some additional restrictive conditions, requiring in particular the equilibrium set to be one-dimensional, we obtain an asymptotic convergence result. Finally, some open problems are listed.
This work was partially supported by the French-Chilean research cooperation program ECOS/CONICYT C04E03. The research was partly realized while the second author was visiting the first one at the CMM, Chile.
The first author was supported by Fondecyt 1020610, Fondap en Matemáticas Aplicadas and Programa Iniciativa Científica Milenio.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Alvarez. On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control Optim., 38:1102–1119, 2000.
V. Arnold. Equations Différentielles Ordinaires. Editions de Moscou, 1974.
H. Attouch and R. Cominetti. A dynamical approach to convex minimization coupling approximation with the steepest descent method. J. Differential Equations 128(2):519–540, 1996.
H. Attouch and M.-O. Czarnecki. Asymptotic control and stabilization of nonlinear oscillators with non isolated equilibria. J. Differential Equations 179:278–310, 2002.
H. Attouch, X. Goudou, and P. Redont. The heavy ball with friction method. I The continuous dynamical system. Commun. Contemp. Math. 2(1):1–34, 2000.
V. Barbu and T. Precupanu. Convexity and Optimization in Banach Spaces. 2nd ed., D. Reidel, Dordrecht, Boston, 1986.
H. Brézis. Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies, No. 5., North-Holland Publishing Co., Amsterdam-London, 1973.
R.E. Bruck. Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J. Funct. Anal. 18:15–26, 1975.
A. Cabot. Inertial gradient-like dynamical system controlled by a stabilizing term, J. Optim. Theory Appl. 120:275–303, 2004.
A. Cabot. The steepest descent dynamical system with control. Applications to constrained minimization. ESAIM Control Optim. Calc. Var., 10:243–258, 2004.
A. Cabot and M.-O. Czarnecki. Asymptotic control of pairs of oscillators coupled by a repulsion, with non isolated equilibria. SIAM J. Control Optim. 41(4):1254–1280, 2002.
M.-O. Czarnecki. Asymptotic control of pairs of oscillators coupled by a repulsion, with non isolated equilibria II: the singular case. SIAM J. Control Optim. 42(6):2145–2171, 2004
W. Hirsch and S. Smale. Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974.
J.P. Lasalle and S. Lefschetz. Stability by Lyapounov’s Direct Method with Applications. Academic Press, New York, 1961.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Alvarez, F., Cabot, A. (2006). On the Asymptotic Behavior of a System of Steepest Descent Equations Coupled by a Vanishing Mutual Repulsion. In: Seeger, A. (eds) Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol 563. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28258-0_1
Download citation
DOI: https://doi.org/10.1007/3-540-28258-0_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28257-0
Online ISBN: 978-3-540-28258-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)