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Viability Theory pp 315–350Cite as

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Lyapunov Functions

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Part of the book series: Systems & Control: Foundations & Applications ((MBC))

Abstract

Consider a differential inclusion x′ ∈ F(x), a function V: X ↦ R + ⋃ {+∞} and a real-valued function ω(·).

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References

  1. TALLOS P. (to appear) Viability problems for nonautonomous differential inclusions,SIAM J. Opt. C.

    Google Scholar 

  2. DEIMLING K. (1989) Extremal solutions of multivalued differential equations II,Preprint (Results in Math)

    Google Scholar 

  3. KANNAI K., TALLOS P. (to appear) Viable trajectories of nonconvex differential inclusions,Preprint

    Google Scholar 

  4. WAZEWSKI T. (1947) Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations diférentielles ordinaires,Ann. Soc. Polon. Math.,20, 279–313

    Google Scholar 

  5. WAZEWSKI T. (1954) Sur une méthode topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles,Proc. of the I.C.M., 132–139

    Google Scholar 

  6. WAZEWSKI T. (1955) Sur la structure de l’ensemble engendré par les intégrales non asymptotiques des équations différentielles,Bull. Acad. Polon. Sc. 3, 143–148

    Google Scholar 

  7. BIELECKI A., KLUCZNY C. (1960) Sur un théorème concernant des systèmes d’équations différentielles ordinaires, Ann. Univ. Mariae Curie-Sklodowska, 14, 115–123

    Google Scholar 

  8. KLUCZNY C. (1962) Sur certaines familles de courbes en relation avec la théorie des différentielles ordinaires II,Ann. Univ. Mariae Curie-Sklodowska, 16, 5–18

    Google Scholar 

  9. BARBASHIN E.A. (1961) On construction of Lyapunov functions for nonlinear systems, (In russian), Proc. 1st Intern. IFAC Congress: Theory of continuous systems-special mathematical problems, Moscow: Acad. Sci. U.S.S.R., 742–750

    Google Scholar 

  10. BARBASHIN E.A. (1970) LYAPUNOV FUNCTIONS, (In rus-sian), Moscow: Nauka

    Google Scholar 

  11. ROXIN E. (1965) On generalized dynamical systems defined by contingent equations,J. Differ. Eq., 1, 188–205

    Google Scholar 

  12. FRANKOWSKA H. (1987) Local controllability and infinitesimal generators of semigroups of set-valued maps,SIAM J. on Control and Optimization, 25, 412–432

    Google Scholar 

  13. KRASOVSKI N. N., SUBBOTIN A. I. (1988) GAME-THEORETICAL CONTROL PROBLEMS, Springer-Verlag

    Google Scholar 

  14. GUSEINOV H. G., SUBBOTIN A. I., USHAKOV V. N. (1985) Derivatives for multivalued mappings with applications to game theoretical problems of control, Problems of Control and Information Theory, Vol. 14, 155–167

    Google Scholar 

  15. KRASOVSKII N.N. (1963) STABILITY OF MOTION, Stanford: Stanford University press

    Google Scholar 

  16. KRASOVSKI N. N., SUBBOTIN A. I. (1974) POSITIONAL DIFFERENTIAL GAMES, Nauka, Moscow

    Google Scholar 

  17. KRASOVSKI N. N. (1986) THE CONTROL OF A DYNAMIC SYSTEM, Nauka, Moscow

    Google Scholar 

  18. KURZHANSKI A.B. (1985) On the analytical description of the viable solutions of a controlled system, Uspekhi Mat. Nauk, 4

    Google Scholar 

  19. KURZHANSKI A.B. (1988) On stochastic filtering approximations of estimation problems for systems with uncertainty,Stochastics, 23, 109–230

    Google Scholar 

  20. KURZHANSKI A.B. (1989) Identification: a theory of guaranteed estimates, in DATA TO MODELS, Kurzhanski, Willems, Eds, Springer-Verlag

    Google Scholar 

  21. TOLSTONOGOV A.A. (1986) DIFFERENTIAL INCLUSIONS IN BANACH SPACES, Nauka, in Russian

    Google Scholar 

  22. PANASYUK A.I., PANASYUK V.I. (1980) An equation generated by a differential inclusion, Mathematical Notes, 27, # 3, 213–218

    Google Scholar 

  23. PANASYUK A.I. (1985) Differential equation of nonconvex attainability sets,Mathematical Notes, 37, # 5, 395–400

    Google Scholar 

  24. PANASYUK A.I. (1986) Dynamics equations for attainable sets in problems of optimization and control under uncertainly, Journal of applied mathematics and mechanics, 50 # 4, 408417

    Google Scholar 

  25. KURZHANSKI A.B. (1977) CONTROL AND OBSERVATION UNDER CONDITIONS OF UNCERTAINTY, Nauka

    Google Scholar 

  26. VALYI I. (1986) Ellipsoidal approximations in problems of control,in MODELLING AMD ADAPTIVE CONTROL, Byrnes-Kurzhanski Eds., Vol. 105, Springer-Verlag

    Google Scholar 

  27. KURZHANSKI A.B., VALYI I. (1988) Set-valued solutions to control problems and their approximations,in ANALYSIS AND OPTIMIZATION OF SYSTEMS, Bensoussan-Lions J.-L, Eds, Vol. 111, 775–785 Springer-Verlag

    Google Scholar 

  28. KURZHANSKI A.B.,VALYI I. (to appear) ELLIPSOIDAL-VALUED DYNAMICS FOR ESTIMATION AND CONTROL

    Google Scholar 

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Authors and Affiliations

  1. EDOMADE (Ecole Doctorale de Mathématique de la Décision), Université de Paris-Dauphine, F-75775, Paris cedex 16, France

    Jean-Pierre Aubin

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  1. Jean-Pierre Aubin
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Aubin, JP. (2009). Lyapunov Functions. In: Viability Theory. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4910-4_11

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