Systèmes dynamiques dissipatifs et applications. (Dissipative dynamical systems and applications). (French) Zbl 0726.58001

Haraux, Alain

Recherches en Mathématiques Appliquées 17. Paris etc.: Masson (ISBN 2-225-82283-2). xi, 129 p. (1991).

Inspired by the classical stability results of Lyapunov and La Salle- Lefschetz, the author and others over the years developed a theory within the framework of functional analysis that deals with corresponding questions concerning such partial differential equations as the wave equation with linear dissipation and the heat equation. This booklet presents a uniform treatment of the stability theory of dynamical systems applying to nonlinear evolution equations in infinite dimensions as well as ordinary differential equations in finite dimensions.
The chapters are headed as follows. 0: Introduction and preliminaries. 1: Basic notions. 2: The invariance principle and applications. 3: Some stability considerations. 4: Partial dissipation and almost periodic trajectories. 5: Attractors of autonomous systems. 6: Non autonomous systems. 7: Stability of non autonomous systems. 8: Attractors of non autonomous systems. 9: Bibliographical comments.

Reviewer: D.Erle (Dortmund)

MSC:

58-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
37C70	Attractors and repellers of smooth dynamical systems and their topological structure
37C75	Stability theory for smooth dynamical systems
34D20	Stability of solutions to ordinary differential equations
35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations
35L20	Initial-boundary value problems for second-order hyperbolic equations

Keywords:

Lyapunov function; invariance principle; almost periodic function; semigroup of operators; attractor; process; ultimate boundedness; stability theory of dynamical systems; evolution equations