Inertial Gradient-Like Dynamical System Controlled by a Stabilizing Term

A. Cabot
Additional contact information
A. Cabot: University of Limoges

Journal of Optimization Theory and Applications, 2004, vol. 120, issue 2, No 3, 275-303

Abstract: 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 C 1 function.

Keywords: Dissipative dynamical systems; nonlinear oscillators; optimization; convex minimization; heavy ball with friction (search for similar items in EconPapers)
Date: 2004
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1023/B:JOTA.0000015685.21638.8d Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:120:y:2004:i:2:d:10.1023_b:jota.0000015685.21638.8d

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1023/B:JOTA.0000015685.21638.8d

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().