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Regularity properties of the minimum-time map

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Nonlinear Synthesis

Part of the book series: Progress in Systems and Control Theory ((PSCT,volume 9))

Abstract

The aim of this paper is to give a survey on some known results concerning the regularity properties of the minimum-time map around an equilibrium point of a control system and to discuss the links of these properties with the viscosity solutions of the Hamilton Jacobi Gellman equation. For sake of simplicity let us consider a control system on R n defined by:

$$(\sum )\dot X = f(X,u) \equiv {f_0}(X) + \sum\limits_{i = 1}^m {{u_i}} {f_i}(X),X(0) = {X_0}$$

where the fi’s are C∞ vector fields and the control map u = (u1,...,um) belongs to the class u of the integrable maps with values in the set

$$\Omega = \left\{ {({\omega _1}, \ldots ,{\omega _m}) \in {R^m}:\left| {{\omega _i}} \right| \leqslant 1,i = 1, \cdots ,m} \right\}.$$

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Via Mezzocannone, 8, 80134, Napoli, Italia

    Gianna Stefani

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  1. Gianna Stefani
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Editor information

Editors and Affiliations

  1. Department of Systems Science and Mathematics, Washington University, 63130, St. Louis, MO, USA

    Christopher I. Byrnes (Chairman) (Chairman)

  2. Department of Systems and Decision Sciences, International Institute for Applied Systems Analysis, A-2361, Laxenburg, Austria

    Alexander B. Kurzhansky (Chairman) (Chairman)

  3. Academy of Sciences, USSR

    Alexander B. Kurzhansky (Chairman) (Chairman)

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© 1991 Springer Science+Business Media New York

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Stefani, G. (1991). Regularity properties of the minimum-time map. In: Byrnes, C.I., Kurzhansky, A.B. (eds) Nonlinear Synthesis. Progress in Systems and Control Theory, vol 9. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-2135-5_21

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  • DOI: https://doi.org/10.1007/978-1-4757-2135-5_21

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-0-8176-3484-1

  • Online ISBN: 978-1-4757-2135-5

  • eBook Packages: Springer Book Archive

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