Abstract
Let H be a real Hilbert space and let \(\Phi :H \to \mathbb{R}\) be a \(\mathcal{C}^1\) function that we wish to minimize. For any potential \(U:H \to \mathbb{R}\) and any control function \(\varepsilon :\mathbb{R}_ + \to \mathbb{R}_ +\) which tends to zero as t→+∞, we study the asymptotic behavior of the trajectories of the following dissipative system:
The (S) system can be viewed as a classical heavy ball with friction equation (Refs. 1–2) plus the control term ε(t)∇U(x(t)). If Φ is convex and ε(t) tends to zero fast enough, each trajectory of (S) converges weakly to some element of argmin Φ. This is a generalization of the Alvarez theorem (Ref. 1). On the other hand, assuming that ε is a slow control and that Φ and U are convex, the (S) trajectories tend to minimize U over argmin Φ when t→+∞. This asymptotic selection property generalizes a result due to Attouch and Czarnecki (Ref. 3) in the case where U(x)=|x|2/2. A large part of our results are stated for the following wider class of systems:
where \(\Psi :\mathbb{R}_ + \times H \to \mathbb{R}\) is a C 1 function.
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References
Alvarez, F., On the Minimizing Property of a Second-Order Dissipative System in Hilbert Space, SIAM Journal on Control and Optimization, Vol. 38, pp. 1102-1119, 2000.
Attouch, H., Goudou, X., and Redont, P., The Heavy Ball with Friction Method, I: The Continuous Dynamical System, Communications in Contemporary Mathematics, Vol. 2, pp. 1-34, 2000.
Attouch, H., and Czarnecki, M. O., Asymptotic Control and Stabilization of Nonlinear Oscillators with Nonisolated Equilibria, Journal of Differential Equations, Vol. 179, pp. 278-310, 2002.
Cabot, A., Motion with Friction of a Heavy Particle on a Manifold: Applications to Optimization, Mathematical Modelling and Numerical Analysis, Vol. 36, pp. 505-516, 2002.
Hale, J. K., Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, Rhode Island, Vol. 25, 1988.
Haraux, A., Fr Systèmes Dynamiques Dissipatifs et Applications, Masson, Paris, France, 1991.
Brézis, H., Asymptotic Behavior of Some Evolution Systems, Nonlinear Evolution Equations, Academic Press, New York, NY, 1978.
Bruck, R. E., Asymptotic Convergence of Nonlinear Contraction Semigroups in Hilbert Space, Journal of Functional Analysis, Vol. 18, pp. 15-26, 1975.
Opial, Z., Weak Convergence of the Sequence of Successive Approximations for Nonexpansive Mappings, Bulletin of the American Mathematical Society, Vol. 73, pp. 591-597, 1967.
Cabot, A., and Czarnecki, M. O., Asymptotic Control of Pairs of Oscillators Coupled by a Repulsion, with Nonisolated Equilibria, SIAM Journal on Control and Optimization, Vol. 41, pp. 1254-1280, 2002.
Attouch, H., and Cominetti, R., A Dynamical Approach to Convex Minimization Coupling Approximation with the Steepest Descent Method, Journal of Differential Equations, Vol. 128, pp. 519-540, 1996.
Tikhonov, A. N., and Arsenine, V., De Méthodes de Résolution de Problèmes Mal Posés, MIR, Moscow, Russia, 1976.
Arnold, V., De Equations Différentielles Ordinaires, Moscow Editions, Moscow, Russia, 1974.
Hirsch, W., and Smale, S., Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, NY, 1974.
Lasalle, J. P., and Lefschetz, S., Stability by Lyapunov Direct Method with Applications, Academic Press, New York, NY, 1961.
Reinhardt, H., Equations Différentielles: Fondements et Applications, 2nd Edition, Dunod, Paris, France, 1989.
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Cabot, A. Inertial Gradient-Like Dynamical System Controlled by a Stabilizing Term. Journal of Optimization Theory and Applications 120, 275–303 (2004). https://doi.org/10.1023/B:JOTA.0000015685.21638.8d
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DOI: https://doi.org/10.1023/B:JOTA.0000015685.21638.8d