Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system

By Alexandre Nolasco de Carvalho, José A. Langa, James C. Robinson, Antonio Suárez. 2007

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In this paper we determine the exact structure of the pullback attractors in non-autonomous problems that are perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria. We show that the pullback attractors of the perturbed systems inherit this structure, and are given as the union of the unstable manifolds of a set of hyperbolic global solutions which are the non-autonomous analogues of the hyperbolic equilibria. We also prove, again parallel to the autonomous case, that all solutions converge as t -> + infinity to one of these hyperbolic global solutions. We then show how to apply these results to systems that are asymptotically autonomous as t -> - infinity) and as t -> + infinity, and use these relatively simple test cases to illustrate a discussion of possible definitions of a forwards attractor in the non-autonomous case. (C) 2007 Elsevier Inc. All rights reserved.

Subfields: Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/5cb61e0fd85b87bdecb7a1fa75be8aea527cb506

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In Control and Systems Engineering

Citations in the corpus

DYNAMICS IN DUMBBELL DOMAINS II. THE LIMITING PROBLEM (2009, background): “In this case, as a consequence of the results in [6], the attractor is characterized”