Abstract
Stability of differential inclusions defined by locally Lipschitz compact valued mappings is considered. It is shown that if such a differential inclusion is globally asymptotically stable, then, in fact, it is uniformly globally asymptotically stable (with respect to initial states in compacts). This statement is trivial for differential equations, but here we provide the extension to compact- (not necessarily convex-) valued differential inclusions. The main result is presented in a context which is useful for control-theoretic applications: a differential inclusion with two outputs is considered, and the result applies to the property of global error detectability.
Similar content being viewed by others
References
F. Albertini and E. D. Sontag, Continuous control-Lyapunov functions for asymptotically controllable time-varying systems. Int. J. Control 72(1999), 1630–1641.
D. Angeli, B. Ingalls, E. D. Sontag, and Y. Wang, Asymptotic characterizations of IOSS. Proc. IEEE Conf. Decision and Control (2001), 881–886.
D. Angeli and E. D. Sontag, Forward completeness, unboundedness observability, and their Lyapunov characterizations. Systems Control Lett. 38(1999), 209–217.
J.-P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, Berlin (1984).
J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990).
F. H. Clarke, Yu. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998).
R. M. Colombo, A. Fryszkowski, T. Rzežuchowski, and V. Staicu, Continuous Selections of Solution Sets of Lipschitzian Differential Inclusions. Funkcial. Ekvac. 34(1991), 321–330.
K. Deimling, Multivalued Differential Equations. Walter De Gruyter, Berlin (1992).
A. F. Filippov, Differential Equations with Discontinuous Right-hand Sides. Kluwer Academic, Dordrecht (1988).
A. Fryszkowski and T. Rzežuchowski, Continuous version of Filippov-Wažewski relaxation theorem. J. Differential Equations 94(1991), 254–265.
B. Ingalls, E. D. Sontag, and Y. Wang, Generalizations of asymptotic gain characterizations of ISS to input-to-output stability. Proc. AACC American Control Conf., Virginia(2001), 2279–2284.
____,An infinite-time relaxation theorem for differential inclusions. Proc. Amer. Math. Soc. 131(2002), 487–499.
A. Isidori, Nonlinear Control Systems. Springer-Verlag, London (1995).
V. Lakshmikantham, S. Lella, and A. A. Martyuk, Practical Stability of Nonlinear Systems. World Scientific, New Jersey (1990).
E. D. Sontag and Y. Wang, New characterizations of the input to state stability property. IEEE Trans. Automat. Control 41(1996), 1283–1294.
A. R. Teel and L. Praly, A smooth Lyapunov function from a class-KLestimate involving two positive semi-definite functions. ESAIM Control Optim. Calc. Var. 5(2000), 313–368.
V. I. Vorotnikov, Stability and stabilization of motion: Research approaches, results, distinctive characteristics. Automat. Remote Control 54(1993), 339–397.
J. Warga, Optimal Control of Differential and Functional Equations.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Angeli, D., Ingalls, B., Sontag, E.D. et al. Uniform Global Asymptotic Stability of Differential Inclusions. Journal of Dynamical and Control Systems 10, 391–412 (2004). https://doi.org/10.1023/B:JODS.0000034437.54937.7f
Issue Date:
DOI: https://doi.org/10.1023/B:JODS.0000034437.54937.7f