A smooth Lyapunov function from a class- 𝒦ℒ estimate involving two positive semidefinite functions

Andrew R. Teel; Laurent Praly

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 5, page 313-367
  • ISSN: 1292-8119

Abstract

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We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class- 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- 𝒦ℒ estimate, exists if and only if the class- 𝒦ℒ estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether all class- 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.

How to cite

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Teel, Andrew R., and Praly, Laurent. "A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimate involving two positive semidefinite functions." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 313-367. <http://eudml.org/doc/197354>.

@article{Teel2010,
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. },
author = {Teel, Andrew R., Praly, Laurent},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Differential inclusions; Lyapunov functions; uniform asymptotic stability.; differential inclusions; uniform asymptotic stability; smooth converse Lyapunov function},
language = {eng},
month = {3},
pages = {313-367},
publisher = {EDP Sciences},
title = {A smooth Lyapunov function from a class-$\{\mathcal\{KL\}\}$ estimate involving two positive semidefinite functions},
url = {http://eudml.org/doc/197354},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Teel, Andrew R.
AU - Praly, Laurent
TI - A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimate involving two positive semidefinite functions
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 5
SP - 313
EP - 367
AB - 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.
LA - eng
KW - Differential inclusions; Lyapunov functions; uniform asymptotic stability.; differential inclusions; uniform asymptotic stability; smooth converse Lyapunov function
UR - http://eudml.org/doc/197354
ER -

References

top
  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
  3. J.-P. Aubin and H. Frankowska, Set-valued Analysis. Birkhauser, Boston (1990).  Zbl0713.49021
  4. A. Bacciotti and L. Rosier, Lyapunov and Lagrange stability: Inverse theorems for discontinuous systems. Math. Control Signals Systems11 (1998) 101-128.  Zbl0919.34051
  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.  
  6. F.H. Clarke, Y.S. Ledyaev and R.J. Stern, Asymptotic stability and smooth Lyapunov functions. J. Differential Equations149 (1998) 69-114.  Zbl0907.34013
  7. F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer (1998).  Zbl1047.49500
  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
  9. W.P. Dayanwansa and C.F. Martin, A converse Lyapunov theorem for a class of dynamical systems which undergo switching. IEEE Trans. Automat. Control44 (1999) 751-764.  
  10. K. Deimling, Multivalued Differential Equations. Walter de Gruyter, Berlin (1992).  Zbl0760.34002
  11. A.F. Filippov, On certain questions in the theory of optimal control. SIAM J. Control1 (1962) 76-84.  Zbl0139.05102
  12. A.F. Filippov, Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers (1988).  
  13. W. Hahn, Stability of Motion. Springer-Verlag (1967).  Zbl0189.38503
  14. F.C. Hoppensteadt, Singular perturbations on the infinite interval. Trans. Amer. Math. Soc.123 (1966) 521-535.  Zbl0151.12502
  15. J. Kurzweil, On the inversion of Ljapunov's second theorem on stability of motion. Amer. Math. Soc. Trans. Ser. 224 (1956) 19-77.  
  16. V. Lakshmikantham, S. Leela and A.A. Martynyuk, Stability Analysis of Nonlinear Systems. Marcel Dekker, Inc. (1989).  Zbl0676.34003
  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
  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
  19. A.M. Lyapunov, The general problem of the stability of motion. Math. Soc. of Kharkov, 1892 (Russian). [English Translation: Internat. J. Control55 (1992) 531-773].  
  20. I.G. Malkin, On the question of the reciprocal of Lyapunov's theorem on asymptotic stability. Prikl. Mat. Mekh.18 (1954) 129-138.  
  21. J.L. Massera, On Liapounoff's conditions of stability. Ann. of Math.50 (1949) 705-721.  Zbl0038.25003
  22. J.L. Massera, Contributions to stability theory. Ann. of Math.64 (1956) 182-206. (Erratum: Ann. of Math.68 (1958) 202.)  Zbl0070.31003
  23. A.M. Meilakhs, Design of stable control systems subject to parametric perturbations. Avtomat. i Telemekh.10 (1978) 5-16.  
  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.  
  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.  
  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
  27. A.A. Movchan, Stability of processes with respect to two measures. Prikl. Mat. Mekh. (1960) 988-1001.  
  28. I.P. Natanson, Theory of Functions of a Real Variable. Vol. 1. Frederick Ungar Publishing Co. (1974).  
  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
  30. E.D. Sontag, Comments on integral variants of ISS. Systems Control Lett.34 (1998) 93-100.  Zbl0902.93062
  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.  Zbl0985.93051
  33. E.D. Sontag and Y. Wang, Lyapunov characterizations of input to output stability. SIAM J. Control Optim. (to appear).  Zbl0968.93076
  34. A.M. Stuart and A.R. Humphries, Dynamical Systems and Numerical Analysis. Cambridge University Press, New York (1996).  Zbl0869.65043
  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
  36. J. Tsinias, A Lyapunov description of stability in control systems. Nonlinear Anal.13 (1989) 63-74.  Zbl0695.93083
  37. J. Tsinias and N. Kalouptsidis, Prolongations and stability analysis via Lyapunov functions of dynamical polysystems. Math. Systems Theory20 (1987) 215-233.  Zbl0642.93052
  38. J. Tsinias, N. Kalouptsidis and A. Bacciotti, Lyapunov functions and stability of dynamical polysystems. Math. Systems Theory19 (1987) 333-354.  Zbl0628.93056
  39. V.I. Vorotnikov, Stability and stabilization of motion: Research approaches, results, distinctive characteristics. Avtomat. i Telemekh. (1993) 3-62.  Zbl0800.93947
  40. F.W. Wilson, Smoothing derivatives of functions and applications. Trans. Amer. Math. Soc.139 (1969) 413-428.  Zbl0175.20203
  41. T. Yoshizawa, Stability Theory by Lyapunov's Second Method. The Mathematical Society of Japan (1966).  Zbl0144.10802
  42. K. Yosida, Functional Analysis, 2nd Edition. Springer Verlag, New York (1968).  Zbl0152.32102

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