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Measure-driven impulsive systems : stabilization, optimal control and applications

University of British Columbia

This dissertation studies various standard facets of nonlinear control problems in the impulsive setting, using a framework of measure-driven systems. Containing a Borel measure in their dynamics, these systems model significant time scale discrepancies; the measure may weight actions at instants, producing trajectories that mix discrete and continuous dynamics on the "fast" and "slow" time scales, respectively. A central feature of our work is the careful use of a time reparametrization to transform these systems into standard, non-impulsive ones, so that the wealth of recent results in nonlinear control may be applied. Closed-loop stabilization of impulsive control systems containing a measure in the dynamics is addressed. It is proved that, as for regular affine systems, an almost everywhere continuous stabilizing impulsive feedback control law exists for such impulsive systems. An example illustrating the loop closing features is also presented. Necessary conditions for optimal control have recently been developed in the non-convex case by Clarke and Vinter, among others. We extend these results to generalized differential inclusions where a signed, vector-valued measure appears. In particular, we offer a set of stratified necessary conditions in optimal control of measure-driven systems, as well as a set of standard (global) conditions under weak regularity hypotheses on the differential inclusion maps. An auxiliary result essential to our proof extends existing free end-time necessary conditions results to Clarke's stratified framework. We work in the context of pseudo-Lipschitz multifunctions, which provide localized Lipschitz-like properties in the absence of convexity. We take a well-evolved solution concept framework in new directions, introducing a workable system of state-dependent measures and measure-based constraints, such as a forced impulse schedule, a restriction to purely discrete impulse dynamics or a state-dependent impulse restriction, and prove necessary conditions in optimal control for this new framework. This is an important step in the renewed use of measure-driven systems in modeling a broad range of applications within a familiar, mathematically sound framework. Taken together, these results span a broad range of topics in nonlinear, state-space control in the impulsive context, and refresh the measure-driven framework, paving the way for future research and further value in applications.

Text

2010-01-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/17450

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/