Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T06:07:45.821Z Has data issue: false hasContentIssue false
  • English
  • Français

A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimateinvolving two positive semidefinite functions

Published online by Cambridge University Press:  15 August 2002

Andrew R. Teel  and
Laurent Praly
Andrew R. Teel
Affiliation:
ECE Department, University of California, Santa Barbara, CA 93106, U.S.A.; [email protected].
Laurent Praly
Affiliation:
Centre Automatique et Systèmes, École des Mines de Paris, 35 rue Saint Honoré, 77305 Fontainebleau Cedex, France; [email protected].
Get access

Abstract

We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class-${\mathcal{KL}}$ estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-${\mathcal{KL}}$ estimate, exists if and only if the class-${\mathcal{KL}}$ estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether all class-${\mathcal{KL}}$ estimates are robust. One sufficient condition for robustness is that the original differential inclusion is locally Lipschitz. Another sufficient condition is that the two positive semidefinite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for differential equations and differential inclusions that have appeared in the literature.

Keywords

Differential inclusionsLyapunov functionsuniform asymptotic stability.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 313 - 367
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A.N. Atassi and H.K. Khalil, A separation principle for the control of a class of nonlinear systems, in Proc. of the 37th IEEE Conference on Decision and Control. Tampa, FL (1998) 855-860.
J.-P. Aubin and A. Cellina, Differential Inclusions: Set-valued Maps and Viability Theory. Springer-Verlag, New York (1984).
J.-P. Aubin and H. Frankowska, Set-valued Analysis. Birkhauser, Boston (1990).
Bacciotti, A. and Rosier, L., Lyapunov and Lagrange stability: Inverse theorems for discontinuous systems. Math. Control Signals Systems 11 (1998) 101-128. CrossRef
Barbashin, E.A. and Krasovskii, N.N., On the existence of a function of Lyapunov in the case of asymptotic stability in the large. Prikl. Mat. Mekh. 18 (1954) 345-350.
Clarke, F.H., Ledyaev, Y.S. and Stern, R.J., Asymptotic stability and smooth Lyapunov functions. J. Differential Equations 149 (1998) 69-114. CrossRef
F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer (1998).
Clarke, F.H., Stern, R.J. and Wolenski, P.R., Subgradient criteria for monotonicity, the Lipschitz condition, and convexity. Canad. J. Math. 45 (1993) 1167-1183. CrossRef
Dayanwansa, W.P. and Martin, C.F., A converse Lyapunov theorem for a class of dynamical systems which undergo switching. IEEE Trans. Automat. Control 44 (1999) 751-764. CrossRef
K. Deimling, Multivalued Differential Equations. Walter de Gruyter, Berlin (1992).
Filippov, A.F., On certain questions in the theory of optimal control. SIAM J. Control 1 (1962) 76-84.
A.F. Filippov, Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers (1988).
W. Hahn, Stability of Motion. Springer-Verlag (1967).
Hoppensteadt, F.C., Singular perturbations on the infinite interval. Trans. Amer. Math. Soc. 123 (1966) 521-535. CrossRef
Kurzweil, J., On the inversion of Ljapunov's second theorem on stability of motion. Amer. Math. Soc. Trans. Ser. 2 24 (1956) 19-77.
V. Lakshmikantham, S. Leela and A.A. Martynyuk, Stability Analysis of Nonlinear Systems. Marcel Dekker, Inc. (1989).
Lakshmikantham, V. and Salvadori, L., Massera, On type converse theorem in terms of two different measures. Bull. U.M.I. 13 (1976) 293-301.
Lin, Y., Sontag, E.D. and Wang, Y., A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim. 34 (1996) 124-160. CrossRef
A.M. Lyapunov, The general problem of the stability of motion. Math. Soc. of Kharkov, 1892 (Russian). [English Translation: Internat. J. Control 55 (1992) 531-773].
Malkin, I.G., On the question of the reciprocal of Lyapunov's theorem on asymptotic stability. Prikl. Mat. Mekh. 18 (1954) 129-138.
Massera, J.L., Liapounoff's, On conditions of stability. Ann. of Math. 50 (1949) 705-721. CrossRef
Massera, J.L., Contributions to stability theory. Ann. of Math. 64 (1956) 182-206. (Erratum: Ann. of Math. 68 (1958) 202.) CrossRef
Meilakhs, A.M., Design of stable control systems subject to parametric perturbations. Avtomat. i Telemekh. 10 (1978) 5-16.
A.P. Molchanov and E.S. Pyatnitskii, Lyapunov functions that specify necessary and sufficient conditions of absolute stability of nonlinear nonstationary control systems I. Avtomat. i Telemekh. (1986) 63-73.
A.P. Molchanov and E.S. Pyatnitskiin, Lyapunov functions that specify necessary and sufficient conditions of absolute stability of nonlinear nonstationary control systems II. Avtomat. i Telemekh. (1986) 5-14.
Molchanov, A.P. and Pyatnitskii, E.S., Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Systems Control Lett. 13 (1989) 59-64. CrossRef
A.A. Movchan, Stability of processes with respect to two measures. Prikl. Mat. Mekh. (1960) 988-1001.
I.P. Natanson, Theory of Functions of a Real Variable. Vol. 1. Frederick Ungar Publishing Co. (1974).
E.P. Ryan, Discontinuous feedback and universal adaptive stabilization, in Control of Uncertain Systems, edited by D. Hinrichsen and B. Martensson. Birkhauser, Boston (1990) 245-258.
Sontag, E.D., Comments on integral variants of ISS. Systems Control Lett. 34 (1998) 93-100. CrossRef
E.D. Sontag and Y. Wang, A notion of input to output stability, in Proc. European Control Conf. Brussels (1997), Paper WE-E A2, CD-ROM file ECC958.pdf.
Sontag, E.D. and Wang, Y., Notions of input to output stability. Systems Control Lett. 38 (1999) 235-248. CrossRef
E.D. Sontag and Y. Wang, Lyapunov characterizations of input to output stability. SIAM J. Control Optim. (to appear).
A.M. Stuart and A.R. Humphries, Dynamical Systems and Numerical Analysis. Cambridge University Press, New York (1996).
Teel, A.R. and Praly, L., Tools for semiglobal stabilization by partial state and output feedback. SIAM J. Control Optim. 33 (1995) 1443-1488. CrossRef
Tsinias, J., Lyapunov de, Ascription of stability in control systems. Nonlinear Anal. 13 (1989) 63-74. CrossRef
Tsinias, J. and Kalouptsidis, N., Prolongations and stability analysis via Lyapunov functions of dynamical polysystems. Math. Systems Theory 20 (1987) 215-233. CrossRef
Tsinias, J., Kalouptsidis, N. and Bacciotti, A., Lyapunov functions and stability of dynamical polysystems. Math. Systems Theory 19 (1987) 333-354. CrossRef
V.I. Vorotnikov, Stability and stabilization of motion: Research approaches, results, distinctive characteristics. Avtomat. i Telemekh. (1993) 3-62.
Wilson, F.W., Smoothing derivatives of functions and applications. Trans. Amer. Math. Soc. 139 (1969) 413-428. CrossRef
T. Yoshizawa, Stability Theory by Lyapunov's Second Method. The Mathematical Society of Japan (1966).
K. Yosida, Functional Analysis, 2nd Edition. Springer Verlag, New York (1968).