×

zbMATH — the first resource for mathematics

A smooth Lyapunov function from a class-\(\mathcal{KL}\) estimate involving two positive semidefinite functions. (English) Zbl 0953.34042
Summary: The authors consider differential inclusions where a positive semidefinite function of the solutions satisfies a class-\({\mathcal K\mathcal L}\) estimate in terms of time and a second positive semidefinite function of the initial condition. They show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-\(\mathcal K\mathcal L\) estimate, exists if and only if the class-\(\mathcal K\mathcal L\) estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains the open question whether all class-\(\mathcal K\mathcal L\) 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.

MSC:
34D20 Stability of solutions to ordinary differential equations
34A60 Ordinary differential inclusions
PDF BibTeX XML Cite
Full Text: DOI Link EuDML
References:
[1] 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.
[2] J.-P. Aubin and A. Cellina, Differential Inclusions: Set-valued Maps and Viability Theory. Springer-Verlag, New York ( 1984). Zbl0538.34007 MR755330 · Zbl 0538.34007
[3] J.-P. Aubin and H. Frankowska, Set-valued Analysis. Birkhauser, Boston ( 1990). Zbl0713.49021 MR1048347 · Zbl 0713.49021
[4] A. Bacciotti and L. Rosier, Lyapunov and Lagrange stability: Inverse theorems for discontinuous systems. Math. Control Signals Systems 11 ( 1998) 101-128. Zbl0919.34051 MR1628047 · Zbl 0919.34051 · doi:10.1007/BF02741887
[5] E.A. Barbashin and N.N Krasovskii, On the existence of a function of Lyapunov in the case of asymptotic stability in the large. Prikl. Mat. Mekh. 18 ( 1954) 345-350. Zbl0055.32005 MR62301 · Zbl 0055.32005
[6] F.H. Clarke, Y.S. Ledyaev and R.J. Stern, Asymptotic stability and smooth Lyapunov functions. J. Differential Equations 149 ( 1998) 69-114. Zbl0907.34013 MR1643670 · Zbl 0907.34013 · doi:10.1006/jdeq.1998.3476
[7] F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer ( 1998). Zbl1047.49500 MR1488695 · Zbl 1047.49500
[8] F.H. Clarke, R.J. Stern and P.R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition, and convexity. Canad. J. Math. 45 ( 1993) 1167-1183. Zbl0810.49016 MR1247540 · Zbl 0810.49016 · doi:10.4153/CJM-1993-065-x
[9] W.P. Dayanwansa and C.F. Martin, A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE Trans. Automat. Control 44 ( 1999) 751-764. Zbl0960.93046 MR1684429 · Zbl 0960.93046 · doi:10.1109/9.754812
[10] K. Deimling, Multivalued Differential Equations. Walter de Gruyter, Berlin ( 1992). Zbl0760.34002 MR1189795 · Zbl 0760.34002
[11] A.F. Filippov, On certain questions in the theory of optimal control. SIAM J. Control 1 ( 1962) 76-84. Zbl0139.05102 MR149985 · Zbl 0139.05102 · doi:10.1137/0301006
[12] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers ( 1988). Zbl0664.34001 MR1028776 · Zbl 0664.34001
[13] W. Hahn, Stability of Motion. Springer-Verlag ( 1967). Zbl0189.38503 MR223668 · Zbl 0189.38503
[14] F.C. Hoppensteadt, Singular perturbations on the infinite interval. Trans. Amer. Math. Soc. 123 ( 1966) 521-535. Zbl0151.12502 MR194693 · Zbl 0151.12502 · doi:10.2307/1994672
[15] J. Kurzweil, On the inversion of Ljapunov’s second theorem on stability of motion. Amer. Math. Soc. Trans. Ser. 2 24 ( 1956) 19-77. Zbl0127.30703 · Zbl 0127.30703 · eudml:11835
[16] V. Lakshmikantham, S. Leela and A.A. Martynyuk, Stability Analysis of Nonlinear Systems. Marcel Dekker, Inc. ( 1989). Zbl0676.34003 MR984861 · Zbl 0676.34003
[17] V. Lakshmikantham and L. Salvadori, On Massera type converse theorem in terms of two different measures. Bull. U.M.I. 13 ( 1976) 293-301. Zbl0351.34030 MR440136 · Zbl 0351.34030
[18] Y. Lin, E.D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim. 34 ( 1996) 124-160. Zbl0856.93070 MR1372908 · Zbl 0856.93070 · doi:10.1137/S0363012993259981
[19] 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]. Zbl0786.70001 MR1154209 · Zbl 0786.70001
[20] I.G. Malkin, On the question of the reciprocal of Lyapunov’s theorem on asymptotic stability. Prikl. Mat. Mekh. 18 ( 1954) 129-138. Zbl0055.32004 · Zbl 0055.32004
[21] J.L. Massera, On Liapounoff’s conditions of stability. Ann. of Math. 50 ( 1949) 705-721. Zbl0038.25003 MR35354 · Zbl 0038.25003 · doi:10.2307/1969558
[22] J.L. Massera, Contributions to stability theory. Ann. of Math. 64 ( 1956) 182-206. (Erratum: Ann. of Math. 68 ( 1958) 202.) Zbl0070.31003 MR79179 · Zbl 0070.31003 · doi:10.2307/1969955
[23] A.M. Meilakhs, Design of stable control systems subject to parametric perturbations. Avtomat. i Telemekh. 10 ( 1978) 5-16. Zbl0419.93038 MR533365 · Zbl 0419.93038
[24] 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. Zbl0607.93039 MR839959 · Zbl 0607.93039
[25] 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. Zbl0618.93051 MR848396 · Zbl 0618.93051
[26] A.P. Molchanov and E.S. Pyatnitskii, Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Systems Control Lett. 13 ( 1989) 59-64. Zbl0684.93065 MR1006848 · Zbl 0684.93065 · doi:10.1016/0167-6911(89)90021-2
[27] A.A. Movchan, Stability of processes with respect to two measures. Prikl. Mat. Mekh. ( 1960) 988-1001. Zbl0100.08401 · Zbl 0100.08401 · doi:10.1016/0021-8928(60)90004-6
[28] I.P. Natanson, Theory of Functions of a Real Variable. Vol. 1. Frederick Ungar Publishing Co. ( 1974). Zbl0064.29102 MR67952 · Zbl 0064.29102
[29] 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. Zbl0726.93069 MR1206689 · Zbl 0726.93069
[30] E.D. Sontag, Comments on integral variants of ISS. Systems Control Lett. 34 ( 1998) 93-100. Zbl0902.93062 MR1629012 · Zbl 0902.93062 · doi:10.1016/S0167-6911(98)00003-6
[31] 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. · Zbl 0901.93062
[32] E.D. Sontag and Y. Wang, Notions of input to output stability. Systems Control Lett. 38 ( 1999) 235-248. Zbl0985.93051 MR1754906 · Zbl 0985.93051 · doi:10.1016/S0167-6911(99)00070-5
[33] E.D. Sontag and Y. Wang, Lyapunov characterizations of input to output stability. SIAM J. Control Optim. (to appear). Zbl0968.93076 MR1339057 · Zbl 0968.93076 · doi:10.1137/S0363012999350213
[34] A.M. Stuart and A.R. Humphries, Dynamical Systems and Numerical Analysis. Cambridge University Press, New York ( 1996). Zbl0869.65043 MR1402909 · Zbl 0869.65043
[35] A.R. Teel and L. Praly, Tools for semiglobal stabilization by partial state and output feedback. SIAM J. Control Optim. 33 ( 1995) 1443-1488. Zbl0843.93057 MR1348117 · Zbl 0843.93057 · doi:10.1137/S0363012992241430
[36] J. Tsinias, A Lyapunov description of stability in control systems. Nonlinear Anal. 13 ( 1989) 63-74. Zbl0695.93083 MR973369 · Zbl 0695.93083 · doi:10.1016/0362-546X(89)90035-7
[37] J. Tsinias and N. Kalouptsidis, Prolongations and stability analysis via Lyapunov functions of dynamical polysystems. Math. Systems Theory 20 ( 1987) 215-233. Zbl0642.93052 MR920776 · Zbl 0642.93052 · doi:10.1007/BF01692066
[38] J. Tsinias, N. Kalouptsidis and A. Bacciotti, Lyapunov functions and stability of dynamical polysystems. Math. Systems Theory 19 ( 1987) 333-354. Zbl0628.93056 MR888495 · Zbl 0628.93056 · doi:10.1007/BF01704919
[39] V.I. Vorotnikov, Stability and stabilization of motion: Research approaches, results, distinctive characteristics. Avtomat. i Telemekh. ( 1993) 3-62. Zbl0800.93947 MR1225444 · Zbl 0800.93947
[40] F.W. Wilson, Smoothing derivatives of functions and applications. Trans. Amer. Math. Soc. 139 ( 1969) 413-428. Zbl0175.20203 MR251747 · Zbl 0175.20203 · doi:10.2307/1995333
[41] T. Yoshizawa, Stability Theory by Lyapunov’s Second Method. The Mathematical Society of Japan ( 1966). Zbl0144.10802 MR208086 · Zbl 0144.10802
[42] K. Yosida, Functional Analysis, 2nd Edition. Springer Verlag, New York ( 1968). Zbl0435.46002 MR239384 · Zbl 0435.46002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.