Skip to main content

Viability tubes

  • Conference paper
  • First Online:
Modelling and Adaptive Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 105))

Abstract

We define viability tubes and invarlant tubes of a differential inclusion, we study some asymptotic properties and we characterize them by showing that the indicator functions of their graphs are solutions to the contingent Hamilton-Jacobi equation. We provide some examples of viability tubes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aubin, J.-P. and A. Cellina (1984) Differential Inclusions, Springer-Verlag (Grundlehren der Math. Wissenschaften, Vol. 264, pp. 1–342)

    Google Scholar 

  • Aubin, J.-P. and F.H. Clarke (1977) Monotone Invariant Solutions to differential Inclusions, J. London Math. Soc., 16, pp. 357–366

    Google Scholar 

  • Aubin, J.-P. and I. Ekeland, (1984) Applied Nonlinear Analysis, Wiley-Interscience

    Google Scholar 

  • Aubin, J.-P., H. Frankowska and C. Olech (1986) Controllability of convex processes, SIAM J. of Control and Optimization

    Google Scholar 

  • Clarke, F.H. (1983) Optimization and Nonsmooth Analysis, Wiley-Interscience

    Google Scholar 

  • Clarke, F.H. and R.B. Vinter (1983) Local Optimality Conditions and Lipschitzian Solutions to the Hamilton-Jacobi Equation, SIAM J. of Control and Optimization, 21(6), pp. 865–870

    Google Scholar 

  • Clarke, F.H. and R.B. Vinter (1986) On the Relationship between the Dynamic Programming and the Maximum Principle, Preprint CRM, Université de Montréal

    Google Scholar 

  • Crandall, M.G. and P.L. Lions (1983) Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math Soc. 277, pp. 1–42

    Google Scholar 

  • Crandall, M.G., L.C. Evans and P.L. Lions (1984) Some Properties of Viscosity Solutions of Hamilton-Jacobi Equation, Trans. Amer. Math. Soc., 282 (2), pp. 487–502

    Google Scholar 

  • Frankowska, H. (1986a) Contingent Hamilton-Jacobi Equations, IIASA, WP-86-

    Google Scholar 

  • Frankowska, H. (1986b) Contingent Cones to Reachable Sets of Control Systems, Preprint CRM-1381, Université de Montréal

    Google Scholar 

  • Frankowska, H. (1986c) Local Controllability of Control Systems with Feedback. Preprint CRM-1364, Université de Montréal

    Google Scholar 

  • Frankowska, H. (1986d) Local Controllability and Infinitesimal Generators of Semigroups of Set-valued Maps, SIAM J. of Control and Optimization

    Google Scholar 

  • Frankowska, H. (1986e) The Maximum Principle for the Differential Inclusions with End Point Constraints, SIAM J. of Control and Optimization

    Google Scholar 

  • Haddad, G. and Lasry J.-M. (1983) Periodic Solutions of Functional Differential Inclusions and Fixed Points of S-Selectionable Correspondences, J. Math. Anal. Appl.

    Google Scholar 

  • Haddad, G. (1981) Monotone Trajectories of Differential with Memory, Israel J. Math s. 39, pp. 38–100

    Google Scholar 

  • Kurzhanski, A.B. and T.F. Filippova (1986) On Viable Solutions for Uncertain Systems, IIASA, CP-86-11

    Google Scholar 

  • Kurzhanski, A.B. (1977) Control and Observation under Conditions of Uncertainty, Nauka (in Russian)

    Google Scholar 

  • Kurzhanski, A.B. (1986) On the Analytical Description of the Viable Solutions of a Controlled System, Uspekhi Mat. Nauk. 4

    Google Scholar 

  • Kurzhanski, A.B. (1986) On the Solution Sets for Uncertain Systems with Phase Constraints, IIASA, WP-86-11

    Google Scholar 

  • Lions, P.-L. (1982) Generalized Solutions of Hamilton-Jacobi Equations, Pitman

    Google Scholar 

  • Rockafellar, R.T. (1967) Monotone Processes of Convex and Concave Type, Mem. of AMS #77

    Google Scholar 

  • Rockafellar, R.T. (1970) Convex Analysis, Princton University Press

    Google Scholar 

  • Rockafellar, R.T. (1979) La théorie des sous-gradients, Presses de l'Université de Montréal

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Christopher Ian Byrnes Alexander B. Kurzhanski

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag

About this paper

Cite this paper

Aubin, J.P. (1988). Viability tubes. In: Byrnes, C.I., Kurzhanski, A.B. (eds) Modelling and Adaptive Control. Lecture Notes in Control and Information Sciences, vol 105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0043175

Download citation

  • DOI: https://doi.org/10.1007/BFb0043175

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-19019-6

  • Online ISBN: 978-3-540-38904-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics