Article Contents
Article Contents

# Classical converse theorems in Lyapunov's second method

• Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems. This technique makes use of an auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific system without the need to generate system solutions. An important question is the converse or reversability of Lyapunov's second method; i.e., given a specific stability property does there exist an appropriate Lyapunov function? We survey some of the available answers to this question.
Mathematics Subject Classification: Primary: 93D05, 93D30, 93D20; Secondary: 93D10.

 Citation:

•  [1] B. D. O. Anderson, Stability of control systems with multiple nonlinearities, Journal of the Franklin Institute, 282 (1966), 155-160.doi: 10.1016/0016-0032(66)90317-6. [2] B. D. O. Anderson and J. B. Moore, New results in linear system stability, SIAM Journal on Control, 7 (1969), 398-414.doi: 10.1137/0307029. [3] B. D. O. Anderson and J. B. Moore, Detectability and stabilizability of time-varying discrete-time linear systems, SIAM Journal on Control and Optimization, 19 (1981), 20-32.doi: 10.1137/0319002. [4] D. Angeli and E. D. Sontag, Forward completeness, unboundedness observability, and their Lyapunov characterizations, Systems & Control Letters, 38 (1999), 209-217.doi: 10.1016/S0167-6911(99)00055-9. [5] H. Antosiewicz, A survey of Lyapunov's second method, Contributions to Nonlinear Oscillations, Princeton University Press, 1958, 147-166. [6] T. M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, USA, 1957. [7] L. Arnold and B. Schmalfuss, Lyapunov's second method for random dynamical systems, Journal of Differential Equations, 177 (2001), 235-265.doi: 10.1006/jdeq.2000.3991. [8] A. Bacciotti and L. Rosier, Liapunov and Lagrange stability: Inverse theorems for discontinuous systems, Mathematics of Control, Signals and Systems, 11 (1998), 101-128.doi: 10.1007/BF02741887. [9] E. A. Barbashin, On the theory of general dynamical systems, (Russian) Ucen. Zap. Moskov. Gos. Univ., 135 (1948), 110-133. [10] E. A. Barbashin, Existence of smooth solutions of some linear equations with partial derivatives, Doklady Akademii Nauk SSSR, 72 (1950), 445-447. [11] E. A. Barbashin and N. N. Krasovskii, On the stability of motion in the large, (Russian) Doklady Akademii Nauk SSSR, 86 (1952), 453-456. [12] E. A. Barbashin and N. N. Krasovskii, On the existence of a function of Lyapunov in the case of asymptotic stability in the large, (Russian) Prikladnaya Matematika i Mekhanika, 18 (1954), 345-350. [13] R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (eds. R. W. Brockett, R. S. Millman and H. J. Sussman), Progr. Math., 27, Birkhäuser Boston, Boston, MA, 1983, 181-191. [14] C. Cai, A. R. Teel and R. Goebel, Smooth Lyapunov functions for hybrid systems, Part I: Existence is equivalent to robustness, IEEE Transactions on Automatic Control, 52 (2007), 1264-1277.doi: 10.1109/TAC.2007.900829. [15] C. Cai, A. R. Teel and R. Goebel, Smooth Lyapunov functions for hybrid systems, Part II: (Pre-)asymptotically stable compact sets, IEEE Transactions on Automatic Control, 53 (2007), 734-748.doi: 10.1109/TAC.2008.919257. [16] N. G. Chetayev, The Stability of Motion, Pergamon Press, 1961; Translated from the 2nd Edition in Russian of 1956. [17] F. H. Clarke, Y. S. Ledyaev, L. Rifford and R. J. Stern, Feedback stabilization and Lyapunov functions, SIAM Journal on Control and Optimization, 39 (2000), 25-48.doi: 10.1137/S0363012999352297. [18] F. H. Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Transactions on Automatic Control, 42 (1997), 1394-1407.doi: 10.1109/9.633828. [19] F. H. Clarke, Y. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions, Journal of Differential Equations, 149 (1998), 69-114.doi: 10.1006/jdeq.1998.3476. [20] F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998. [21] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series no. 38, American Mathematical Society, 1978. [22] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd edition, Wiley-Interscience, 2006. [23] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, 1992.doi: 10.1515/9783110874228. [24] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, 1988.doi: 10.1007/978-94-015-7793-9. [25] P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete and Continuous Dynamical Systems, Series B, 20 (2015). [26] R. Goebel, R. G. Sanfelice and A. R. Teel, Hybrid Dynamical Systems: Modeling, Stability, and Robustness, Princeton University Press, 2012. [27] S. P. Gordon, On converses to the stability theorems for difference equations, SIAM Journal on Control, 10 (1972), 76-81.doi: 10.1137/0310007. [28] L. Grüne, F. Camilli and F. Wirth, A generalization of Zubov's method to perturbed systems, SIAM Journal on Control and Optimization, 40 (2001), 496-515.doi: 10.1137/S036301299936316X. [29] L. Grüne, P. E. Kloeden, S. Siegmund and F. R. Wirth, Lyapunov's second method for nonautonomous differential equations, Discrete and Continuous Dynamical Systems, 18 (2007), 375-403.doi: 10.3934/dcds.2007.18.375. [30] L. Grüne and O. S. Serea, Differential games and Zubov's method, SIAM Journal on Control and Optimization, 49 (2011), 2349-2377.doi: 10.1137/100787829. [31] W. Hahn, Theory and Application of Liapunov's Direct Method, Prentice-Hall, 1963; Translated from the German Edition of 1959. [32] W. Hahn, Stability of Motion, Springer-Verlag, 1967. [33] B. E. Hitz and B. D. O. Anderson, Discrete positive-real functions and their application to system stability, Proceedings of the Institution of Electrical Engineers, 116 (1969), 153-155.doi: 10.1049/piee.1969.0031. [34] F. C. Hoppensteadt, Singular perturbations on the infinite interval, Transactions of the American Mathematical Society, 123 (1966), 521-535.doi: 10.1090/S0002-9947-1966-0194693-9. [35] B. P. Ingalls, E. D. Sontag and Y. Wang, Measurement to error stability: A notion of partial detectability for nonlinear systems, in Proceedings of the 41st IEEE Conference on Decision and Control, Vol. 4, Las Vegas, Nevada, USA, 2002, 3946-3951.doi: 10.1109/CDC.2002.1184983. [36] Z.-P. Jiang and Y. Wang, A converse Lyapunov theorem for discrete-time systems with disturbances, Systems & Control Letters, 45 (2002), 49-58.doi: 10.1016/S0167-6911(01)00164-5. [37] R. E. Kalman, Lyapunov function for the problem of Lur'e in automatic control, Proc. Nat. Acad. Sci. U.S.A., 49 (1963), 201-205.doi: 10.1073/pnas.49.2.201. [38] R. E. Kalman and J. E. Bertram, Control system analysis and design via the "second method'' of Lyapunov, Part I, continuous-time systems, Transactions of the AMSE, Series D: Journal of Basic Engineering, 82 (1960), 371-393.doi: 10.1115/1.3662604. [39] R. E. Kalman and J. E. Bertram, Control system analysis and design via the "second method'' of Lyapunov, Part II, discrete-time systems, Transactions of the AMSE, Series D: Journal of Basic Engineering, 82 (1960), 394-400.doi: 10.1115/1.3662605. [40] I. Karafyllis, Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis, IMA Journal of Mathematical Control and Information, 23 (2006), 11-41.doi: 10.1093/imamci/dni037. [41] I. Karafyllis and J. Tsinias, A converse Lyapunov theorem for nonuniform in time global asymptotic stability and its application to feedback stabilization, SIAM Journal on Control and Optimization, 42 (2003), 936-965.doi: 10.1137/S0363012901392967. [42] C. M. Kellett, A compendium of comparison function results, Mathematics of Controls, Signals and Systems, 26 (2014), 339-374.doi: 10.1007/s00498-014-0128-8. [43] C. M. Kellett and A. R. Teel, A converse Lyapunov theorem for weak uniform asymptotic stability of sets, in Proceedings of Mathematical Theory of Networks and Systems, Perpignan, France, 2000. [44] C. M. Kellett and A. R. Teel, Uniform asymptotic controllability to a set implies locally Lipschitz control-Lyapunov function, in Proceedings of the 39th IEEE Conference on Decision and Control, Vol. 4, Sydney, Australia, 2000, 3994-3999.doi: 10.1109/CDC.2000.912339. [45] C. M. Kellett and A. R. Teel, Discrete-time asymptotic controllability implies smooth control-Lyapunov function, Systems & Control Letters, 52 (2004), 349-359.doi: 10.1016/j.sysconle.2004.02.011. [46] C. M. Kellett and A. R. Teel, Weak converse Lyapunov theorems and control Lyapunov functions, SIAM Journal on Control and Optimization, 42 (2004), 1934-1959.doi: 10.1137/S0363012901398186. [47] C. M. Kellett and A. R. Teel, On the robustness of $\mathcal{KL}$-stability for difference inclusions: Smooth discrete-time Lyapunov functions, SIAM Journal on Control and Optimization, 44 (2005), 777-800.doi: 10.1137/S0363012903435862. [48] C. M. Kellett and A. R. Teel, Sufficient conditions for robustness of $\mathcal{KL}$-stability for difference inclusions, Mathematics of Control, Signals and Systems, 19 (2007), 183-205.doi: 10.1007/s00498-007-0016-6. [49] H. K. Khalil, Nonlinear Systems, 2nd edition, Prentice Hall, 1996. [50] R. Khasminskii, Stochastic Stability of Differential Equations, 2nd edition, Springer, 2012.doi: 10.1007/978-3-642-23280-0. [51] P. E. Kloeden, General control systems, in Mathematical Control Theory 1977: Proceedings (ed. W. A. Coppel), Springer-Verlag, Canberra, Australia, 1978, 119-137. [52] P. E. Kloeden, Lyapunov functions for cocycle attractors in nonautonomous difference equations, Izvetsiya Akad Nauk Rep Moldovia Mathematika, 26 (1998), 32-42. [53] P. E. Kloeden, A Lyapunov function for pullback attractors of nonautonomous differential equations, Electronic Journal of Differential Equations Conference 05, (2000), 91-102. [54] P. Kokotović and M. Arcak, Constructive nonlinear control: A historical perspective, Automatica, 37 (2001), 637-662.doi: 10.1016/S0005-1098(01)00002-4. [55] N. N. Krasovskii, On the inversion of the theorems of A. M. Lyapunov and N. G. Chetaev concerning instability for stationary systems of differential equations, (Russian) Prikladnaya Matematika i Mekhanika, 18 (1954), 513-532. [56] N. N. Krasovskii, On the converse of K. P. Persidskii's theorem on uniform stability, (Russian) Prikladnaya Matematika i Mekhanika, 19 (1955), 273-278. [57] N. N. Krasovskii, Transformation of the theorem of A. M. Lyapunov's second method and questions of first-order stability of motion, (Russian) Prikladnaya Matematika i Mekhanika, 20 (1956), 255-265. [58] N. N. Krasovskii, Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay, Stanford University Press, 1963; Translated from the Russian Edition of 1959. [59] M. Krichman, E. D. Sontag and Y. Wang, Input-output-to-state stability, SIAM Journal on Control and Optimization, 39 (2001), 1874-1928.doi: 10.1137/S0363012999365352. [60] M. Krstić, I. Kanellakopoulos and P. Kokotović, Nonlinear and Adaptive Control Design, John Wiley and Sons, Inc., 1995. [61] J. Kurzweil, Transformation of Lyapunov's first theorem on stability of motion, (Russian) Czechoslovak Mathematical Journal, 5 (1955), 382-398. [62] J. Kurzweil, On the inversion of Ljapunov's second theorem on stability of motion, (Russian) Czechoslovak Mathematical Journal, 81 (1956), 217-259, 455-484; English translation in American Mathematical Society Translations (2), 24, 19-77. [63] J. Kurzweil and I. Vrkoč, Transformation of Lyapunov's theorems on stability and Persidskii's theorems on uniform stability, (Russian) Czechoslovak Mathematical Journal, 7 (1957), 254-272. [64] H. J. Kushner, Converse theorems for stochastic Liapunov functions, SIAM Journal on Control and Optimization, 5 (1967), 228-233.doi: 10.1137/0305015. [65] J. La Salle and S. Lefschetz, Stability by Liapunov's Direct Method with Applications, Academic Press, 1961. [66] V. Lakshmikantham and L. Salvadori, On Massera type converse theorem in terms of two different measures, Bollettino dell'Unione Matematica Italiana, 13 (1976), 293-301. [67] A. L. Letov, Stability in Nonlinear Control Systems, Princeton University Press, Princeton, New Jersey, 1961; Translated from the Russian Edition of 1955. [68] Y. Lin, E. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM Journal on Control and Optimization, 34 (1996), 124-160.doi: 10.1137/S0363012993259981. [69] A. I. Lur'e, Some Non-Linear Problems in the Theory of Automatic Control, Her Majesty's Stationery Office, 1957; Translated from the Russian Edition of 1951. [70] A. I. Lur'e and V. N. Postnikov, Stability theory of regulating systems, (Russian) Prikladnaya Matematika i Mekhanika, 8 (1944), 246-248. [71] A. M. Lyapunov, The general problem of the stability of motion, (Russian) Math. Soc. of Kharkov; English Translation, International Journal of Control, 55 (1992), 531-773.doi: 10.1080/00207179208934253. [72] I. G. Malkin, Questions concerning transformation of Lyapunov's theorem on asymptotic stability, (Russian) Prikladnaya Matematika i Mekhanika, 18 (1954), 129-138. [73] I. G. Malkin, Some Problems in the Theory of Nonlinear Oscillations, United States Atomic Energy Commission, 1959, Translated from the Russian Edition of 1956. [74] J. L. Massera, On Liapounoff's conditions of stability, Annals of Mathematics, 50 (1949), 705-721.doi: 10.2307/1969558. [75] J. L. Massera, Contributions to stability theory, Annals of Mathematics, 64 (1956), 182-206; Erratum: Annals of Mathematics, 68 (1958), 202.doi: 10.2307/1969955. [76] A. M. Meilakhs, Design of stable control systems subject to parametric perturbation, Automation and Remote Control, 39 (1979), 1409-1418. [77] A. N. Michel, L. Hou and D. Liu, Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems, Birkhäuser, 2008. [78] A. P. Molchanov and E. S. Pyatnitskii, Lyapunov functions that specifiy necessary and sufficient conditions of absolute stability of nonlinear nonstationary control systems I, Automation and Remote Control, 47 (1986), 344-354. [79] A. P. Molchanov and E. S. Pyatnitskii, Lyapunov functions that specifiy necessary and sufficient conditions of absolute stability of nonlinear nonstationary control systems II, Automation and Remote Control, 47 (1986), 443-451. [80] A. P. Molchanov and E. S. Pyatnitskii, Lyapunov functions that specifiy necessary and sufficient conditions of absolute stability of nonlinear nonstationary control systems III, Automation and Remote Control, 47 (1986), 620-630. [81] A. P. Molchanov and Y. S. Pyatnitskiy, Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Systems & Control Letters, 13 (1989), 59-64.doi: 10.1016/0167-6911(89)90021-2. [82] A. A. Movchan, Stability of processes with respect to two metrics, Journal of Applied Mathematics and Mechanics, 24 (1960), 1506-1524.doi: 10.1016/0021-8928(60)90004-6. [83] M. Patrao, Existence of complete Lyapunov functions for semiflows on separable metric spaces, Far East Journal of Dynamical Systems, 17 (2011), 49-54. [84] K. P. Persidskii, On a theorem of Liapunov, C. R. (Dokl.) Acad. Sci. URSS, 14 (1937), 541-543. [85] V. M. Popov, Absolute stability of nonlinear systems of automatic control, Automation and Remote Control, 22 (1961), 857-875. [86] V. M. Popov, Proprietati de stabilitate si de optimalitate pentru sistemele automate cu mai multe functii de comanda, (Romanian) Studii si Cercetari de Energetica, Academici RPR, 14 (1964), 913-949. [87] V. M. Popov, Hyperstability of Control Systems, Springer-Verlag, 1973, Translated from the Romanian Edition of 1966. [88] A. Rantzer, A dual to Lyapunov's stability theorem, Systems & Control Letters, 42 (2001), 161-168.doi: 10.1016/S0167-6911(00)00087-6. [89] A. Rantzer, An converse theorem for density functions, in Proceedings of the 41st IEEE Conference on Decision and Control, Vol. 2, Las Vegas, Nevada, USA, 2002, 1890-1891.doi: 10.1109/CDC.2002.1184801. [90] L. Rifford, Existence of Lipschitz and semiconcave control-Lyapunov functions, SIAM Journal on Control and Optimization, 39 (2000), 1043-1064.doi: 10.1137/S0363012999356039. [91] L. Rifford, Semiconcave control-Lyapunov functions and stabilizing feedbacks, SIAM Journal on Control and Optimization, 41 (2002), 659-681.doi: 10.1137/S0363012900375342. [92] L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, Systems & Control Letters, 19 (1992), 467-473.doi: 10.1016/0167-6911(92)90078-7. [93] N. Rouche, P. Habets and M. Laloy, Stability Theory by Liapunov's Direct Method, Springer-Verlag, 1977. [94] E. Roxin, On generalized dynamical systems defined by contingent equations, Journal of Differential Equations, 1 (1965), 188-205.doi: 10.1016/0022-0396(65)90019-7. [95] E. Roxin, Stability in general control systems, Journal of Differential Equations, 1 (1965), 115-150.doi: 10.1016/0022-0396(65)90015-X. [96] E. Roxin, On asymptotic stability in control systems, Rendiconti del Circolo Matematico di Palermo, 15 (1966), 193-208.doi: 10.1007/BF02849435. [97] E. Roxin, On stability in control systems, SIAM Journal on Control, 3 (1966), 357-372.doi: 10.1137/0303024. [98] C. E. Shannon and W. Weaver, The Mathematical Theory of Communication, University of Illinois Press, 1949. [99] R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Review, 49 (2007), 545-592.doi: 10.1137/05063516X. [100] D. D. Šiljak, Nonlinear Systems: The Parameter Analysis and Design, John Wiley & Sons Inc., 1969. [101] G. V. Smirnov, Weak asymptotic stability of differential inclusions I, Automation and Remote Control, 51 (1990), 901-908. [102] G. V. Smirnov, Weak asymptotic stability of differential inclusions II, Automation and Remote Control, 51 (1990), 1052-1058. [103] E. D. Sontag, A Lyapunov-like characterization of asymptotic controllability, SIAM Journal on Control and Optimization, 21 (1983), 462-471.doi: 10.1137/0321028. [104] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34 (1989), 435-443.doi: 10.1109/9.28018. [105] E. D. Sontag, Clocks and insensitivity to small measurement errors, ESAIM: Control, Optimization, and the Calculus of Variations, 4 (1999), 537-557.doi: 10.1051/cocv:1999121. [106] E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Systems & Control Letters, 24 (1995), 351-359.doi: 10.1016/0167-6911(94)00050-6. [107] P. Stein, Some general theorems on iterants, Journal of Research of the National Bureau of Standards, 48 (1952), 82-83.doi: 10.6028/jres.048.010. [108] A. Subbaraman and A. R. Teel, A converse Lyapunov theorem for strong global recurrence, Automatica, 49 (2013), 2963-2974.doi: 10.1016/j.automatica.2013.07.001. [109] A. R. Teel, J. P. Hespanha and A. Subbaraman, A converse Lyapunov theorem and robustness for asymptotic stability in probability, IEEE Transactions on Automatic Control, 59 (2014), 2426-2421.doi: 10.1109/TAC.2014.2322431. [110] A. R. Teel and L. Praly, A smooth Lyapunov function from a class-$\mathcal{KL}$ estimate involving two positive semidefinite functions, ESAIM: Control, Optimization, and the Calculus of Variations, 5 (2000), 313-367.doi: 10.1051/cocv:2000113. [111] Y. Z. Tsypkin, The absolute stability of large-scale nonlinear sampled-data systems, (Russian) Doklady Akademii Nauk SSSR, 145 (1962), 52-55. [112] Y. Z. Tsypkin, Absolute stability of equilibrium positions and of responses in nonlinear, sampled-data automatic systems, Automation and Remote Control, 24 (1963), 1457-1470. [113] V. I. Vorotnikov, Partial stability and control: The state-of-the-art and development prospects, Automation and Remote Control, 66 (2005), 511-561.doi: 10.1007/s10513-005-0099-9. [114] I. Vrkoč, A general theorem of Chetaev, (Russian) Czechoslovak Mathematical Journal, 5 (1955), 451-461. [115] J. C. Willems, Dissipative dynamical systems part I: General theory, Archive for Rational Mechanics and Analysis, 45 (1972), 321-351.doi: 10.1007/BF00276493. [116] J. C. Willems, Dissipative dynamical systems part II: Linear systems with quadratic supply rates, Archive for Rational Mechanics and Analysis, 45 (1972), 352-393.doi: 10.1007/BF00276494. [117] F. W. Wilson, Smoothing derivatives of functions and applications, Transactions of the American Mathematical Society, 139 (1969), 413-428.doi: 10.1090/S0002-9947-1969-0251747-9. [118] V. A. Yakubovich, The solution of certain matrix inequalities in automatic control theory, Doklady Akademii Nauk SSSR, 143 (1962), 1304-1307. [119] T. Yoshizawa, On the stability of solutions of a system of differential equations, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math., 29 (1955), 27-33. [120] T. Yoshizawa, Stability Theory by Liapunov's Second Method, Mathematical Society of Japan, 1966. [121] V. I. Zubov, Methods of A. M. Lyapunov and their Application, P. Noordhoff Ltd, Groningen, The Netherlands, 1964; Translated from the Russian Edition of 1957.
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