A smooth Lyapunov function from a class-𝒦ℒ estimate involving two positive semidefinite functions
ESAIM: Control, Optimisation and Calculus of Variations, Volume 5 (2000), pp. 313-367.
@article{COCV_2000__5__313_0,
     author = {Teel, Andrew R. and Praly, Laurent},
     title = {A smooth {Lyapunov} function from a class-$\mathcal {KL}$ estimate involving two positive semidefinite functions},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {313--367},
     publisher = {EDP-Sciences},
     volume = {5},
     year = {2000},
     mrnumber = {1765429},
     zbl = {0953.34042},
     language = {en},
     url = {http://archive.numdam.org/item/COCV_2000__5__313_0/}
}
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Teel, Andrew R.; Praly, Laurent. A smooth Lyapunov function from a class-$\mathcal {KL}$ estimate involving two positive semidefinite functions. ESAIM: Control, Optimisation and Calculus of Variations, Volume 5 (2000), pp. 313-367. http://archive.numdam.org/item/COCV_2000__5__313_0/

[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). | MR | Zbl

[3] J.-P. Aubin and H. Frankowska, Set-valued Analysis. Birkhauser, Boston ( 1990). | MR | Zbl

[4] A. Bacciotti and L. Rosier, Lyapunov and Lagrange stability: Inverse theorems for discontinuous systems. Math. Control Signals Systems 11 ( 1998) 101-128. | MR | Zbl

[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. | MR | Zbl

[6] F.H. Clarke, Y.S. Ledyaev and R.J. Stern, Asymptotic stability and smooth Lyapunov functions. J. Differential Equations 149 ( 1998) 69-114. | MR | Zbl

[7] F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer ( 1998). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[10] K. Deimling, Multivalued Differential Equations. Walter de Gruyter, Berlin ( 1992). | MR | Zbl

[11] A.F. Filippov, On certain questions in the theory of optimal control. SIAM J. Control 1 ( 1962) 76-84. | MR | Zbl

[12] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers ( 1988). | MR | Zbl

[13] W. Hahn, Stability of Motion. Springer-Verlag ( 1967). | MR | Zbl

[14] F.C. Hoppensteadt, Singular perturbations on the infinite interval. Trans. Amer. Math. Soc. 123 ( 1966) 521-535. | MR | Zbl

[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. | EuDML | Zbl

[16] V. Lakshmikantham, S. Leela and A.A. Martynyuk, Stability Analysis of Nonlinear Systems. Marcel Dekker, Inc. ( 1989). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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]. | MR | Zbl

[20] I.G. Malkin, On the question of the reciprocal of Lyapunov's theorem on asymptotic stability. Prikl. Mat. Mekh. 18 ( 1954) 129-138. | Zbl

[21] J.L. Massera, On Liapounoff's conditions of stability. Ann. of Math. 50 ( 1949) 705-721. | MR | Zbl

[22] J.L. Massera, Contributions to stability theory. Ann. of Math. 64 ( 1956) 182-206. (Erratum: Ann. of Math. 68 ( 1958) 202.) | MR | Zbl

[23] A.M. Meilakhs, Design of stable control systems subject to parametric perturbations. Avtomat. i Telemekh. 10 ( 1978) 5-16. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[27] A.A. Movchan, Stability of processes with respect to two measures. Prikl. Mat. Mekh. ( 1960) 988-1001. | Zbl

[28] I.P. Natanson, Theory of Functions of a Real Variable. Vol. 1. Frederick Ungar Publishing Co. ( 1974). | MR | Zbl

[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. | MR | Zbl

[30] E.D. Sontag, Comments on integral variants of ISS. Systems Control Lett. 34 ( 1998) 93-100. | MR | Zbl

[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.

[32] E.D. Sontag and Y. Wang, Notions of input to output stability. Systems Control Lett. 38 ( 1999) 235-248. | MR | Zbl

[33] E.D. Sontag and Y. Wang, Lyapunov characterizations of input to output stability. SIAM J. Control Optim. (to appear). | MR | Zbl

[34] A.M. Stuart and A.R. Humphries, Dynamical Systems and Numerical Analysis. Cambridge University Press, New York ( 1996). | MR | Zbl

[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. | MR | Zbl

[36] J. Tsinias, A Lyapunov description of stability in control systems. Nonlinear Anal. 13 ( 1989) 63-74. | MR | Zbl

[37] J. Tsinias and N. Kalouptsidis, Prolongations and stability analysis via Lyapunov functions of dynamical polysystems. Math. Systems Theory 20 ( 1987) 215-233. | MR | Zbl

[38] J. Tsinias, N. Kalouptsidis and A. Bacciotti, Lyapunov functions and stability of dynamical polysystems. Math. Systems Theory 19 ( 1987) 333-354. | MR | Zbl

[39] V.I. Vorotnikov, Stability and stabilization of motion: Research approaches, results, distinctive characteristics. Avtomat. i Telemekh. ( 1993) 3-62. | MR | Zbl

[40] F.W. Wilson, Smoothing derivatives of functions and applications. Trans. Amer. Math. Soc. 139 ( 1969) 413-428. | MR | Zbl

[41] T. Yoshizawa, Stability Theory by Lyapunov's Second Method. The Mathematical Society of Japan ( 1966). | MR | Zbl

[42] K. Yosida, Functional Analysis, 2nd Edition. Springer Verlag, New York ( 1968). | MR | Zbl