Reformulations for Control Systems and Optimization Problems with Impulses

• Main Author: Blanton, Jacob
• Other Authors: Wolenski, Peter
• Format: Others
• Language: en
• Published: LSU 2014
• Subjects: Mathematics
• Online Access: http://etd.lsu.edu/docs/available/etd-01152014-220519/

This dissertation studies two different techniques for analyzing control systems whose dynamics include impulses, or more specifically, are measure-driven. In such systems, the state trajectories will have discontinuities corresponding to the atoms of the Borel measure driving the dynamics, and these discontinuities require further definition in order for the control system to be treated with the broad range of results available to non-impulsive systems. Both techniques considered involve a reparameterization of the system variables including state, time, and controls. The first method is that of the graph completion, which provides an explicit reparameterization of the time and state variables. The reparameterization is continuous, which allows for the analysis of the system within classical control theory, yet it retains enough information about the discontinuous, or impulsive, trajectories that the results of such analyses may be interpreted for the original impulsive system. We utilize this reparameterization to formulate equivalent solution concepts between impulsive differential inclusions and impulsive differential equations. We also demonstrate that the graph completion is generally equivalent to a solution concept established for a neural spiking model, and make use of a specific such model as a numerical example. The second method considered is similar to the graph completion but differs in that it utilizes implicit reparameterizations of all variables considered as families of functions which meet continuity and other requirements. This is particularly beneficial to optimal control problems as the choices of controls, impulsive and non-impulsive, may be varied within the optimization problem and analysis thereof. Necessary conditions for optimal control problems of Mayer form with fixed end time have been established under this reparameterization technique, and we extend these necessary conditions in a general context to a Mayer problem with free end time. Corollary to this, we deduce necessary conditions for a Bolza problem and a minimum time problem for impulsive control systems. Much of these results are obtained through reformulation techniques.