Homotopia Causal de Trajetorias de Sistemas de Controle
Tez Türü: Doktora
Tezin Yürütüldüğü Kurum: Universidade Estadual de Campinas (UNICAMP), Instituto de Ciencias Matematicas e de Computaçao, Departamento de Matematica, Brezilya
Tez Danışmanı: Luiz San Martin
Tezin Onay Tarihi: 2003
Tezin Dili: Portekizce
Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
Desteklendiği Program: Diğer
Özet:
In this work, we deal with monotonic homotopy, an appropriate variant of homotopy, of trajectories of non-linear control systems as well as monotonic curves in Lie semigroups. We first introduce a concept of regularity for control functions which may be viewed, through a reparametrization, as geralization of normal controls, and consider monotonic homotopy of regular trajectories of a given control system Σ on a manifold M. Then, we show that the set Γ(Σ; x) of monotonic homotopy classes of regular trajectories of Σ starting at a given fixed point x has a differentiable manifold structure
with the same dimension as M. In this connection, Theorem 3.1.1 is one of the major achievements of the thesis. As a consequence of this theorem we get a local diffeomorphism and lift Σ to the manifold Γ(Σ; x) obtaining a system in Γ(Σ; x).
To consider universal properties of Γ(Σ; x), we take a manifold N that covers the accessible set A(Σ,x) via a surjective local diffeomorphism. Comparing the trajectories of the lifted systems on these two manifolds, we construct a map from Γ(Σ; x) into N. This construction is only a mild imitation of the classical theory. We then compare monotonic homotopy with usual homotopy. In particular, we exibit an example of a system admitting trajectories which are homotopic but not monotonically homotopic. We also try to relate our constructions and results to one of the problems presented in [ ] for semigroups in general. We define a fundamental semigroup for monotonic homotopy as an analogue of fundamental group of a topological space. Finally, we particularize the results obtained so far to the context of control sets where the initial value problem that appears throughout the work may be improved assuming x∈intA(x).