NDLI: Set-valued differentials and a nonsmooth version of Chow's theorem

Content Provider	IEEE Xplore Digital Library
Author	Rampazzo, F. Sussmann, H.J.
Copyright Year	2001
Description	Author affiliation: Dipt. di Matematica Pura e Applicata, Padova Univ., Italy (Rampazzo, F.)
Abstract	We present a simple application of a method developed by one of the authors to prove general smooth, nonsmooth, high-order, and hybrid versions of the maximum principle (MP) for finite-dimensional, deterministic optimal control problems without state space constraints. In the classical approach, one constructs "packets of needle variations", linearly approximates these packets at the base value of the variation parameter, and propagates the resulting linear approximations to the terminal point of the trajectory by means of the differentials of the reference flow maps. This technique must be modified by (a) replacing the classical differential by other objects, called generalized differentials ( GDs), (b) replacing the time-varying vector fields that occur in the classical MP by flows, and (c) replacing the needle variations by abstract variations. A notion of CD will yield a version of the MP provided it satisfies some natural properties such as the chain rule and an appropriate "directional open mapping property". Here we show how to apply the general theory to derive a "nonsmooth" version of Chow's theorem, in which some of the vector fields involved are just continuous, while the others are Lipschitz continuous but not necessarily of class C/sup 1/.
Sponsorship	IEEE Control Syst. Soc
Starting Page	2613
Ending Page	2618
File Size	591536
Page Count	6
File Format	PDF
ISBN	0780370619
DOI	10.1109/CDC.2001.980661
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2001-12-04
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Needles Linear approximation Optimal control State-space methods Books Trajectory Vectors Controllability Mathematics
Content Type	Text
Resource Type	Article

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