Extremality, Controllability, and Abundant Subsets of Generalized Control Systems

B. Kaskosz
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B. Kaskosz: University of Rhode Island

Journal of Optimization Theory and Applications, 1999, vol. 101, issue 1, No 5, 73-108

Abstract: Abstract The generalized control system that we consider in this paper is a collection of vector fields, which are measurable in the time variable and Lipschitzian in the state variable. For such system, we define the concept of an abundant subset. Our definition follows the definition of an abundant set of control functions introduced by Warga. We prove a controllability–extremality theorem for generalized control systems, which says, in essence, that either a given trajectory satisfies a type of maximum principle or a neighborhood of the endpoint of the trajectory can be covered by trajectories of an abundant subset. We apply the theorem to a control system in the classical formulation and obtain a controllability–extremality result, which is stronger, in some respects, than all previous results of this type. Finally, we apply the theorem to differential inclusions and obtain, as an easy corollary, a Pontryagin-type maximum principle for nonconvex inclusions.

Keywords: Generalized control systems; nonsmooth control systems; abundant sets of controls; differential inclusions; local controllability; maximum principle (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1023/A:1021719027140

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