Carlos J. Braga Barros

Chain transitive sets for flows on flag bundles

Abstract We study the chain transitive sets and Morse decompositions of flows on fiber bundles whose fibers are compact homogeneous spaces of Lie groups. The emphasis is put on generalized flag manifolds of semi-simple (and reductive) Lie groups. In this case an algebraic description of the chain transitive sets is given. Our approach consists in shadowing the flow by semigroups of homeomorphisms to take advantage of the good properties of the semigroup actions on flag manifolds. The description of the chain components in the flag bundles generalizes a theorem of Selgrade for projective bundles with an independent proof.

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.

On the number of control sets on projective spaces

The purpose of this paper is to provide an upper bound for the number of control sets for linear semigroups acting on a projective space RPd−1. These semigroups and control sets were studied by Colonius and Kliemann (1993) who proved that there are at most d control sets. Here we apply the results of San Martin and Tonelli (1995) about control sets for semigroups in semisimple Lie groups and make a case by case analysis according to the transitive groups on RPd−1 which were classified by Boothby and Wilson (1975, 1979) in order to improve that upper bound. It turns out that in some cases there are at most d/2 or d/4 control sets.

Controllability of discrete-time control system on the symplectic group

Abstract Let x n +1 =exp( A + uB ) x n be a discrete-time control system on R 2n with A and B matrices in the symplectic Lie algebra sp (n, R ) . This paper gives sufficient conditions for the global controllability of the system. These conditions are similar to those of Martin (Math. Control Signal Systems 8 (1995) 279).

Attraction and Lyapunov stability for control systems on vector bundles

Abstract Let π : E → B be a finite-dimensional vector bundle whose base space is compact. In this paper, we study attraction and Lyapunov stability for control systems on E . 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 π : E → B .

Chain control sets and fiber bundles

Let S be a semigroup of homeomorphisms of a compact metric space M and suppose that is a family of subsets of S. This paper gives a characterization of the -chain control sets as intersection of control sets for the semigroups generated by the neighborhoods of the subsets in . We also study the behavior of -chain control sets on principal bundles and their associated bundles.