A dynamical approach for the stability of second order dissipative systems.
type: Item
instance of: scholarly article
title: A dynamical approach for the stability of second order dissipative systems.
zbMATH Open document ID: 1043.34050
published in: Differential and Integral Equations
publication date: 2002-01-01T00:00:00Z
review text: The nonlinear initial value problem \[ u''(t)+\lambda u'(t)+ \nabla\phi(u(t))= 0,\;t\geq 0,\qquad u(0)= u_0,\qquad u'(0)= u_1 \] in a real Hilbert space \(H\) is considered, where \(\phi: H\to\mathbb{R}\). In particular, the question of whether \(u(t)\) approaches a critical point of \(\phi\) as \(t\to\infty\) is addressed. This question is motivated by the problem of numerically minimizing a function \(\phi\). First, global existence and uniqueness are verified under the assumptions that \(\phi\) is continuously differentiable, \(\phi\) is bounded from below and \(\nabla\phi\) is Lipschitz continuous on the bounded subsets of \(H\). In Theorem 3.1, it is proven that \(u(t)\) converges to a minimizer of \(\phi\) weakly and \(\phi(u(t))\) converges to a minimum of \(\phi\) as \(t\to\infty\). Theorem 3.4 proves under the additional assumption \(\phi\) is strongly convex that \(u(t)\) is norm convergent to the unique global minimizer of \(\phi\). Theorem 3.6 demonstrates that Theorem 3.1 generalizes a result due to \textit{R. E. Bruck} [J. Funct. Anal. 18, 15--26 (1975; Zbl 0319.47041)]. Theorem 3.8 shows that under the assumptions that \(\phi\) is a Morse function (a differentiability-type assumption) and trajectories are precompact, \(u(t)\) converges to a critical point of \(\phi\). Finally, Theorem 4.1 demonstrates that if \(H= \mathbb{R}^n\) and if \(\phi\) is analytic, then \(u(t)\) converges to a critical point of \(\phi\).
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 1836169
zbMATH Keywords: critical point
zbMATH Keywords: minimizer
zbMATH Keywords: minimization
zbMATH Keywords: strongly convex
author: Mohammed Aassila
reviewed by: Daniel C. Biles
MaRDI profile type: Publication
Recommended article: THE HEAVY BALL WITH FRICTION METHOD, I. THE CONTINUOUS DYNAMICAL SYSTEM: GLOBAL EXPLORATION OF THE LOCAL MINIMA OF A REAL-VALUED FUNCTION BY ASYMPTOTIC ANALYSIS OF A DISSIPATIVE DYNAMICAL SYSTEM
Recommended article: Periodic solutions of nonautonomous second-order Hamiltonian systems with even-typed potentials
Recommended article: Q3481512
Recommended article: Multiple solutions of some boundary value problems with parameters
Recommended article: Q3807550
Recommended article: Periodic solutions for second order Hamiltonian systems with a change sign potential
Recommended article: The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy
Recommended article: Forced second order conservative systems with periodic nonlinearity
Recommended article: Q3768039
Recommended article: A result on singularly perturbed elliptic problems
description: scientific article; zbMATH DE number 1836169