On stability and controllability for semigroup actions (original) (raw)

Some Aspects of Stability for Semigroup Actions and Control Systems

Journal of Dynamics and Differential Equations, 2014

The present paper introduces both the notions of Lagrange and Poisson stabilities for semigroup actions. Let S be a semigroup acting on a topological space X with mapping σ : S × X → X , and let F be a family of subsets of S. For x ∈ X the motion σ x : S → X is said to be forward Lagrange stable if the orbit Sx has compact closure in X. The point x is forward F-Poisson stable if and only if it belongs to the limit set ω (x, F). The concept of prolongational limit set is also introduced and used to describe nonwandering points. It is shown that a point x is F-nonwandering if and only if x lies in its forward F-prolongational limit set J (x, F). The paper contains applications to control systems.

Fundamental semigroups for local control sets

Annali di Matematica Pura ed Applicata, 2006

In this paper we associate a semigroup to a locally maximal subset of complete controllability, i.e., a local control set. This fundamental semigroup is based on equivalence classes under homotopies in the set of trajectories. It reflects the structure of the set of closed (trajectory) loops in the local control set. We discuss the relations between different local control sets and prove a Van Kampen-type theorem for their unions and intersections.

Points of continuity for semigroup actions

Transactions of the American Mathematical Society, 1984

The purpose of this paper is to provide a more unified approach to questions involving the existence of points of joint continuity in separately continuous semigroup actions by deriving a small number of general principles which suffice to deduce previously derived results and generalizations thereof. The first major result gives sufficient conditions for a point to be a point of joint continuity in a general setting of "migrants", a useful symmetric generalization of semigroup actions. Results concerning actions of semigroups with group-like properties follow. In the latter part of the paper the notion of a subordinate point is introduced and joint continuity at subordinate points for various settings is proved. Finally, these results are applied to linear actions on locally convex spaces.

A note on sensitivity of semigroup actions

Semigroup Forum, 2008

It is well known that for a transitive dynamical system (X, f ) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey [2], Silverman and Glasner and Weiss . In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X, f ) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in the book of Glasner .

On the structural properties of the bounded control set of a linear control system

Nonlinear Differential Equations and Applications NoDEA

The present paper shows that the closure of the bounded control set of a linear control system contains all the bounded orbits of the system. As a consequence, we prove that the closure of this control set is the continuous image of the cartesian product of the set of control functions by the central subgroup associated with the drift of the system.

The topological dynamics of semigroup actions

Transactions of the American Mathematical Society, 2000

In these notes we explore the fine structure of recurrence for semigroup actions, using the algebraic structure of compactifications of the acting semigroup.

Local controllability and semigroups of diffeomorphisms

Acta Applicandae Mathematicae, 1993

The local structure of orbits of semigroups, generated by families of diffeomorphisms, is studied by Lie theory methods. New sufficient conditions for local controllability are obtained which take into account ordinary, as well as fast-switching variations.

On the Classification of Control Sets

Lecture Notes in Control and Information Sciences

The controllability behavior of nonlinear control systems is described by associating semigroups to locally maximal subsets of complete controllability, i.e., local control sets. Periodic trajectories are called equivalent if there is a 'homotopy' between them involving only trajectories. The resulting object is a semigroup, which we call the dynamic index of the local control set. It measures the different ways the system can go through the local control set.

A semigroup associated to a linear control system on a Lie group

Systems & Control Letters

Let us consider a linear control system Σ on a connected Lie group G. It is known that the accessibility set A from the identity e is in general not a semigroup. In this article we associate a new algebraic object SΣ to Σ which turns out to be a semigroup, allowing the use of the semigroup machinery to approach Σ In particular, we obtain some controllability results.