Josiney A. Souza

Complete Lyapunov functions of control systems

Abstract In this paper, the notion of complete Lyapunov function of control systems is introduced. The purpose is to determine a continuous real-valued function that describes the global structure of the system. The existence of complete Lyapunov functions is proved for certain classes of affine control systems on compact manifolds.

On limit behavior of semigroup actions on noncompact spaces

This paper is produced in response to the questioning of Morse decomposition for semigroup actions on noncompact spaces. We show how the limit behavior can be studied in arbitrary topological spaces by using powerful tools such as the Stone-Čech compactification and shadowing semigroups. We extend Conley’s characterization of chain recurrence in terms of attractors from the setting of flows on compact metric spaces to the setting of semigroup actions on any topological space.

On the number of maximal chain transitive sets in fiber bundles

Abstract. This article studies chain transitivity for semigroup actions on fiber bundles whose typical fibers are compact topological spaces. We discuss the number of maximal chain transitive sets and, as a consequence, we obtain conditions for the existence of a finest Morse decomposition. Some of the results obtained are applied to orthonormal frame and Stiefel manifolds. A description of the maximal chain transitive sets is provided in terms of the action of shadowing semigroups. This description is well known in the literature under the hypothesis of local transitivity. Here, we exclude the hypothesis of local transitivity when the state space is a compact quotient space of a topological group.

LEBESGUE COVERING LEMMA ON NONMETRIC SPACES

In this paper the Lebesgue covering lemma is extended from the setting of metric spaces to the setting of admissible spaces. An admissible space is a topological space endowed with an admissible family of open coverings, and need not be metric. The paper contains applications to uniform continuity and dimension theory.

On Morse decompositions of control systems

In this article both notions of Morse decomposition and dynamic Morse decomposition of control systems are studied. The main objective is to present the basic conditions which assure that a Morse decomposition is a dynamic Morse decomposition, and vice-versa.

On chain recurrence for bitransformation semigroups

Let (S, X, T) be a bitransformation semigroup. The purpose of this article is to describe the chain recurrence of (S, X) in terms of the action of T. It includes an exposition on chain transitivity of skew-product flows associated to differential systems on Lie groups.

On stability and controllability for semigroup actions

This paper deals with stability and controllability for semigroup actions by using the topological method of admissible family of open coverings. The main results state a relationship of stable sets and control sets. The classical notion of controllability relates to the Poisson stability. The concept of prolongational control set relates to the Lyapunov stability.

A characterization of completely regular spaces

This paper proves that uniform spaces and admissible spaces form the same class of topological spaces. This result characterizes a completely regular space as a topological space that admits an admissible family of open coverings. In addition, the admissible family of coverings provides an interesting methodology of studying aspects of uniformity and dynamics in completely regular spaces.

On the existence of global attractors in principal bundles

ABSTRACT This article discusses necessary and sufficient conditions for the existence of global attractors for transformation semigroups on principal bundles. Since the global attractor is a compact set, the discussion involves the compactness of the fibres. A compact structure group is a necessary condition for the existence of the global attractor. In specific situations, the global attractor exists if the structure group is compact.

Characterization of uniform attractors for control systems

This paper is devoted to the characterisation of uniform attractors for control systems by means of Lyapunov functions. We consider a uniform attractor that is compact and positively invariant by the system. We present the relationship between the concept of uniform attractor and the Conley concept of attractor.