Title page for ETD etd-04112005-120514


Type of Document Dissertation
Author Rios, Vinicio Rafael
Author's Email Address rios@math.lsu.edu
URN etd-04112005-120514
Title Dissipative Lipschitz Dynamics
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Peter R. Wolenski Committee Chair
Frank Neubrander Committee Member
Jimmie Lawson Committee Member
Richard A. Litherland Committee Member
Robert Perlis Committee Member
Marcio S. DeQueiroz Dean's Representative
Keywords
  • minimal time function
  • Hamilton-Jacobi inequality
  • strong invariance
  • differential inclusion
Date of Defense 2005-04-01
Availability unrestricted
Abstract
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.

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