Fundamental semigroups for local 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.

Discrete and Continuous Dynamical Systems, 2005

An algebraic semigroup describing the dynamic behavior is associated to compact, locally maximal chain transitive subsets. The construction is based on perturbations and associated local control sets. The dependence on the perturbation structure is analyzed.

Topological Methods in Nonlinear Analysis, 2016

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.

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.

Journal of Differential Equations, 2005

This paper considers monotonic (or causal) homotopy between trajectories of control systems. The main result is the construction of an analogue of the simply connected covering space. The constructed covering ( , x) has the structure of a manifold and satisfies the property that two trajectories are monotonic homotopic if and only if the end points of their liftings coincide.

Let G be a noncompact semi-simple Lie group, consider S a semi-group which contains a large Lie semigroup. We computer the homo-topy type pn(C), where C is the invariant control set of the homoge-neous space G=P with P Ì G a parabolic subgroup of G.

Journal of Dynamical and Control Systems, 2003

The local controllability behavior near an equilibrium is discussed. If the Jacobian of the linearized system is hyperbolic, uniqueness of local control sets is established.

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 number of examples are considered.

2021

We say that a control system is locally controllable if the attainable set from any state x contains an open neighborhood of x, while it is controllable if the attainable set from any state is the entire state manifold. Quite surprisingly, the question of whether local controllability implies global controllability seems not to have been considered in the literature. We show in this paper that a control system satisfying local controllability is controllable.

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.