Journal of Optimization Theory and Applications

, Volume 120, Issue 2, pp 275–303

Inertial Gradient-Like Dynamical System Controlled by a Stabilizing Term

  • A. Cabot
Article

DOI: 10.1023/B:JOTA.0000015685.21638.8d

Cite this article as:
Cabot, A. Journal of Optimization Theory and Applications (2004) 120: 275. doi:10.1023/B:JOTA.0000015685.21638.8d

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:
$$({\text{S) }}\ddot x(t) + \gamma \dot x(t) + \triangledown \Phi (x(t)) + \varepsilon (t)\triangledown U(x(t)) = 0,{\text{ }}\gamma >{\text{0}}{\text{.}}$$
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:
$$({\text{GS) }}\ddot x(t) + \gamma \dot x(t) + \triangledown _x \Psi (t,x(t)) = 0,$$
where \(\Psi :\mathbb{R}_ + \times H \to \mathbb{R}\) is a C1 function.
Dissipative dynamical systems nonlinear oscillators optimization convex minimization heavy ball with friction 

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • A. Cabot
    • 1
  1. 1.Laboratory LACO, Faculty of ScienceUniversity of LimogesLimogesFrance

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