Dissipative Lipschitz dynamics

In this dissertation we study two related important issues in control theory: invariance of dynamical systems and Hamilton-Jacobi theory associated with optimal control theory. Given a control system modelled as a differential inclusion, we provide necessary and sufficient conditions for the strong invariance property of the system when the dynamic satisfies a dissipative Lipschitz condition. We show that when the dynamic is almost upper semicontinuous and satisfies the dissipative Lipschitz property, these conditions can be expressed in terms of approximate Hamilton-Jacobi inequalities, which subsumes the classic infinitesimal characterization of strongly invariant systems given under the Lipschitz assumtion. In the important case when the dynamic of the system is the sum of a maximal dissipative and a Lipschitz multifunction, the approximate inequalities turn into an exact mixed type inequality that involves the lower and upper Hamiltonian of the dissipative and the Lipschitz piece respectively. We then extend this Hamiltonian characterization to nonautonomous systems by assuming a potentially discontinuous differential inclusion whose right-hand side is the sum of an almost upper semicontinuous dissipative and an almost lower semicontinuous dissipative Lipschitz multifunction. Finally, a Hamilton-Jacobi theory is developed for the minimal time problem of a system with possibly discontinuous monotone Lipschitz dynamic. This is achieved by showing the minimal time function associated to an upper semicontinuous and a monotone Lipschitz data is characterized as the unique proximal semi-solution to an approximate Hamilton-Jacobi equation satisfying an analytical boundary condition.

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Applied Mathematics

Grantor

Mathematics

Year dc:date.available

2005

Author dc:creator

• Rios, Vinicio Rafael

• minimal time function
• Hamilton-Jacobi inequality
• strong invariance
• differential inclusion

Statement dc:rights

• unrestricted
• Release the entire work immediately for access worldwide.

Identifier

etd-04112005-120514
https://repository.lsu.edu/gradschool_dissertations/1404

OAI identifier oai:identifier

oai:repository.lsu.edu:gradschool_dissertations-2403