Carlos J. Braga Barros’s research while affiliated with State University of Maringá and other places

Let be a finite-dimensional vector bundle whose base space is compact. In this paper, we study attraction and Lyapunov stability for control systems on . We prove that, under certain conditions, the concepts of Conley attractor, uniform attractor, attractor, exponential attractor, asymptotically stable set and stable set are equivalent for the zero section of .

The present paper is dedicated to the study of various aspects of attraction and stability for semigroup actions on topological spaces. The main purpose is to present the connections among the distinct notions of attractors and stable sets. The concept of Conley attractor is also investigated and related to the other notions of attractors. All the results are applied to the theory of control systems.

In this paper we develop a theory of Lyapunov stability for generalized flows on principal and associated bundles. We present a study of Lyapunov stability and attraction in the total space of a principal bundle by means of the action of the structure group.We also relate limit sets, prolongations, prolongational limit sets, attracting sets and stable sets in the total space of an associated bundle to the corresponding concepts in the fibers.

Let M and N be admissible Hausdorff topological spaces endowed with admissible families of open coverings. Assume that F is a semigroup acting on both M and N. In this paper we study the behavior of limit sets, prolongations, prolongational limit sets, attracting sets, attractors, and Lyapunov stable sets (all concepts defined for the action of the semigroup S) under equivariant maps and semiconjugations from M to N.

In this paper we introduce a theory of Lyapunov stability of sets for semigroup actions on Tychonoff spaces. We also present the main properties and the main results relating these new concepts. We generalize several concepts and results of Lyapunov stable sets from Bhatia and Hajek (Local Semi-Dynamical Systems. Lecture Notes in Mathematics, vol. 90. Springer, Berlin, 1969), Bhatia and Szegö (Dynamical Systems: Stability Theory and Applications. Lecture Notes in Mathematics, vol. 35. Springer, Berlin, 1967; and Stability Theory of Dynamical Systems. Springer, Berlin, 1970).

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.

In this paper, we introduce the concept of dynamic Morse decomposition for an action of a semigroup of homeomorphisms. Conley has shown in [5, Sec. 7] that the concepts of Morse decomposition and dynamic Morse decompositions are equivalent for flows in metric spaces. Here, we show that a Morse decomposition for an action of a semigroup of homeomorphisms of a compact topological space is a dynamic Morse decomposition. We also define Morse decompositions and dynamic Morse decompositions for control systems on manifolds. Under certain condition, we show that the concept of dynamic Morse decomposition for control system is equivalent to the concept of Morse decomposition.

Let S{\mathcal{S}} be a semigroup acting on a topological space M. We study finest Morse decompositions for the action of S{\mathcal{S}} on M. This concept depends on a family of subsets of S{\mathcal{S}} . For certain semigroups and families it recovers the concept of Morse decomposition for flows and semiflows. This paper also studies the behaviour of Morse decompositions for semigroup actions on principal bundles and their associated bundles. The emphasis is put on the study of those decompositions considering their projections onto the base space and their intersections with the fibers. KeywordsMorse sets-Morse decomposition-Associated bundles-Chain transitivity

The concepts of Morse decompositions and dynamic Morse decompositions are equivalent for flows. In this paper we show that these concepts are not equivalent for Morse decompositions of semigroup of homeomorphisms on topological spaces. We give an example of a dynamic Morse decomposition which is not a Morse decomposition on compactifications of topological spaces. Other examples of Morse decompositions are also provided.

Let S{\mathcal{S}} be a semigroup acting on a topological space M. We define attractors for the action of S{\mathcal{S}} on M. This concept depends on a family F{\mathcal{F}} of subsets of S{\mathcal{S}} . For certain semigroups and families it recovers the concept of attractors for flows or semiflows. We define and study the complementary repeller of an attractor. We also characterize the set of chain recurrent points in terms of attractors. KeywordsLimit sets-Attractors-Complementary repellers-Chain recurrence Mathematics Subject Classification (2000)37B35-37B25