NDLI: Asymptotic Control of Pairs of Oscillators Coupled by a Repulsion, with Nonisolated Equilibria II: The Singular Case

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Czarnecki, Marc-Olivier
Copyright Year	2004
Abstract	Let $\phi : H\rightarrow \mathbb R$ be a $\mathcal{C}^1$ function on a real Hilbert space H, and let $\gamma >0$ be a positive damping parameter. For any (singular) repulsive potential $V:H\setminus\{0\}\to \mathbb R_+$, i.e., such that $\lim_{ z \to 0} V(z)=+\infty$, and any control function $\varepsilon :\mathbb R_+\to \mathbb R_+\setminus\{0\}$ which tends to zero as $t\to +\infty$, we study the asymptotic behavior of the trajectories of the coupled dissipative system of nonlinear oscillators $$ \left \{ \begin{array}{l} \ddot{x}+\gamma \dot{x}+\nabla\phi(x)+\varepsilon(t)\nabla V(x-y)=0,\vspace{1mm}\\ \ddot{y}+\gamma \dot{y}+\nabla\phi(y)-\varepsilon(t)\nabla V(x-y)=0. \end{array} \right. \leqno{({\rm HBFC}^2_{sing})} $$ This system is the singular version of the regular $({\rm HBFC}^2_{reg})$ system studied in [A. Cabot and M.-O. Czarnecki, SIAM J. Control Optim., 41 (2002), pp. 1254--1280], where the potential V is defined on the whole space H. The purpose of this paper is to obtain whenever possible the same existence and convergence results in the singular case as in the regular case considered by A. Cabot and M.-O. Czarnecki. This study is mainly motivated by a better convergence behavior of $({\rm HBFC}^2_{sing})$ with the same sharp condition on the control $\varepsilon$ exhibited in [H. Attouch and M.-O. Czarnecki, J. Differential Equations, 179 (2002), pp. 278--310] and by A. Cabot and M.-O. Czarnecki. Precisely, when $H=\mathbb R$, and if $\varepsilon$ is a "slow" control, i.e., $\int_0^{+\infty} \varepsilon(t)dt=+\infty$, then the trajectories x and y converge to extremal points of the set $S=\{\lambda\in \mathbb R, \nabla\phi(\lambda)=0\}$ of the equilibria of $\phi$. The awkward case in A. Cabot and M.-O. Czarnecki, where the trajectories may have the same limit, disappears. Of importance, from a physical point of view, we thus can consider actual, for example electromagnetic, repulsive potentials.
Starting Page	2145
Ending Page	2171
Page Count	27
File Format	PDF
ISSN	03630129
DOI	10.1137/S036301290342320X
e-ISSN	10957138
Issue Number	6
Volume Number	42
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-26
Access Restriction	Subscribed
Subject Keyword	Control problems Nonlinear equations coupled system slow control singular potential nonlinear oscillator Asymptotic properties Perturbations, asymptotics Dynamical systems in optimization and economics heavy ball with friction global optimization Problems involving ordinary differential equations
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Applied Mathematics

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