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- W4233121385 abstract "Let $phi : Hrightarrow R$ be a $mathcal{C}^1$ function on a real Hilbert space H and let $gamma > 0$ be a positive damping parameter. For any repulsive potential $V:Hto R_+$ and any control function $varepsilon :R_+to R_+$ which tends to zero as $tto +infty$, we study the asymptotic behavior of the trajectories of the coupled dissipative system of nonlinear oscillators $$ left { begin{array}{l} ddot{x}+gamma dot{x}+nablaphi(x)+varepsilon(t)nabla V(x-y)=0, ddot{y}+gamma dot{y}+nablaphi(y)-varepsilon(t)nabla V(x-y)=0. end{array} right. leqno{rm (HBFC^2)} $$ We first provide general existence results and show that $nabla phi (x(t))to 0$ and $nabla phi (y(t))to 0$ when $tto +infty$, assuming either that the trajectory (x,y) is bounded, or that the potential V is bounded and that $phi$ satisfies the following limit condition: vspace*{abovedisplayskip} begin{itemize} item[rm (LIM)] For every sequence $ (z_n)subset H $ such that $ lim_{nto +infty}|z_n|=+infty,$ there exists a subsequence $(z_{varphi(n)})$ such that $$ lim_{n to +infty}{phi}(z_{varphi(n)})=+infty qquad mbox{or} qquad lim_{n to +infty}nabla{phi}(z_{varphi(n)})=0. $$ end{itemize} noindent If $varepsilon(t)$ does not tend to zero too rapidly as $tto +infty$, then the term $varepsilon(t) nabla V(x-y)$ asymptotically repulses the trajectories one from the other. Precisely, when $H=R$, 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 $S={lambdain R, nablaphi(lambda)=0}$ of the equilibria of $phi$ (when $Sne emptyset$), or they have the same limit. In particular, when S is reduced to an interval---for example, if $phi$ is convex---this allows us to obtain a global description of the set S. We provide numerical experiments which make our convergence results more precise." @default.
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- W4233121385 date "2002-01-01" @default.
- W4233121385 modified "2023-09-27" @default.
- W4233121385 title "Asymptotic Control of Pairs of Oscillators Coupled by a Repulsion, with Nonisolated Equilibria I: The Regular Case" @default.
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- W4233121385 doi "https://doi.org/10.1137/s0363012901385198" @default.
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