Abstract :
Let F be a general dynamical system defined on a complete locally compact metric
space X. We give a slightly improved version of the Lyapunov characterization of
asymptotic stability in one of our previous works and provide a short self-contained
proof for the existence of arbitrarily small positively invariant neighborhoods of compact
asymptotically stable sets in the present context. Based on these results, we prove that the
uniform attractors of F are connected and are stable with respect to perturbations under
appropriate conditions. We are also interested in the dynamical properties of differential
inclusion: x (t ) ∈ f (x(t)) on Rm. First, we show that if no solutions of the system blow
up in finite future time, then its reachable mapping F is a general dynamical system.
Then we discuss some asymptotic stability properties of the system. In particular, we
prove that if there exists a nonempty compact connected subset M of Rm such that M
attracts a neighborhood of itself, then the system has a connected uniform attractor A. We
also prove that A is stable with respect to both internal and external perturbations. More
precisely, we prove that when λ > 0 is sufficiently small, the perturbed system: x (t ) ∈ conf (x(t)+λ B1)+λ B1 has a connected uniform attractorAλ; moreover, δH(Aλ,A)→0
as λ→0, where δH(·, ·) is the Hausdorff distance in Rm.
2002 Elsevier Science (USA). All rights reserved.
