American Institute of Mathematical Sciences

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

H. Antosiewicz, A survey of Lyapunov's second method, Contributions to Nonlinear Oscillations, Princeton University Press, 1958, 147-166.  Google Scholar

T. M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, USA, 1957.  Google Scholar

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.  Google Scholar

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.  Google Scholar

E. A. Barbashin, On the theory of general dynamical systems, (Russian) Ucen. Zap. Moskov. Gos. Univ., 135 (1948), 110-133. Google Scholar

E. A. Barbashin, Existence of smooth solutions of some linear equations with partial derivatives, Doklady Akademii Nauk SSSR, 72 (1950), 445-447.  Google Scholar

E. A. Barbashin and N. N. Krasovskii, On the stability of motion in the large, (Russian) Doklady Akademii Nauk SSSR, 86 (1952), 453-456.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

N. G. Chetayev, The Stability of Motion, Pergamon Press, 1961; Translated from the 2nd Edition in Russian of 1956.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.  Google Scholar

C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series no. 38, American Mathematical Society, 1978.  Google Scholar

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd edition, Wiley-Interscience, 2006.  Google Scholar

K. Deimling, Multivalued Differential Equations, Walter de Gruyter, 1992. doi: 10.1515/9783110874228.  Google Scholar

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete and Continuous Dynamical Systems, Series B, 20 (2015). Google Scholar

R. Goebel, R. G. Sanfelice and A. R. Teel, Hybrid Dynamical Systems: Modeling, Stability, and Robustness, Princeton University Press, 2012.  Google Scholar

S. P. Gordon, On converses to the stability theorems for difference equations, SIAM Journal on Control, 10 (1972), 76-81. doi: 10.1137/0310007.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

W. Hahn, Theory and Application of Liapunov's Direct Method, Prentice-Hall, 1963; Translated from the German Edition of 1959.  Google Scholar

W. Hahn, Stability of Motion, Springer-Verlag, 1967.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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. Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

H. K. Khalil, Nonlinear Systems, 2nd edition, Prentice Hall, 1996. Google Scholar

R. Khasminskii, Stochastic Stability of Differential Equations, 2nd edition, Springer, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar

P. E. Kloeden, General control systems, in Mathematical Control Theory 1977: Proceedings (ed. W. A. Coppel), Springer-Verlag, Canberra, Australia, 1978, 119-137.  Google Scholar

P. E. Kloeden, Lyapunov functions for cocycle attractors in nonautonomous difference equations, Izvetsiya Akad Nauk Rep Moldovia Mathematika, 26 (1998), 32-42.  Google Scholar

P. E. Kloeden, A Lyapunov function for pullback attractors of nonautonomous differential equations, Electronic Journal of Differential Equations Conference 05, (2000), 91-102.  Google Scholar

P. Kokotović and M. Arcak, Constructive nonlinear control: A historical perspective, Automatica, 37 (2001), 637-662. doi: 10.1016/S0005-1098(01)00002-4.  Google Scholar

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.  Google Scholar

N. N. Krasovskii, On the converse of K. P. Persidskii's theorem on uniform stability, (Russian) Prikladnaya Matematika i Mekhanika, 19 (1955), 273-278.  Google Scholar

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. Google Scholar

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.  Google Scholar

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.  Google Scholar

M. Krstić, I. Kanellakopoulos and P. Kokotović, Nonlinear and Adaptive Control Design, John Wiley and Sons, Inc., 1995. Google Scholar

J. Kurzweil, Transformation of Lyapunov's first theorem on stability of motion, (Russian) Czechoslovak Mathematical Journal, 5 (1955), 382-398.  Google Scholar

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. Google Scholar

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.  Google Scholar

H. J. Kushner, Converse theorems for stochastic Liapunov functions, SIAM Journal on Control and Optimization, 5 (1967), 228-233. doi: 10.1137/0305015.  Google Scholar

J. La Salle and S. Lefschetz, Stability by Liapunov's Direct Method with Applications, Academic Press, 1961.  Google Scholar

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.  Google Scholar

A. L. Letov, Stability in Nonlinear Control Systems, Princeton University Press, Princeton, New Jersey, 1961; Translated from the Russian Edition of 1955.  Google Scholar

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.  Google Scholar

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. Google Scholar

A. I. Lur'e and V. N. Postnikov, Stability theory of regulating systems, (Russian) Prikladnaya Matematika i Mekhanika, 8 (1944), 246-248.  Google Scholar

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.  Google Scholar

I. G. Malkin, Questions concerning transformation of Lyapunov's theorem on asymptotic stability, (Russian) Prikladnaya Matematika i Mekhanika, 18 (1954), 129-138. Google Scholar

I. G. Malkin, Some Problems in the Theory of Nonlinear Oscillations, United States Atomic Energy Commission, 1959, Translated from the Russian Edition of 1956. Google Scholar

J. L. Massera, On Liapounoff's conditions of stability, Annals of Mathematics, 50 (1949), 705-721. doi: 10.2307/1969558.  Google Scholar

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.  Google Scholar

A. M. Meilakhs, Design of stable control systems subject to parametric perturbation, Automation and Remote Control, 39 (1979), 1409-1418.  Google Scholar

A. N. Michel, L. Hou and D. Liu, Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems, Birkhäuser, 2008.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

M. Patrao, Existence of complete Lyapunov functions for semiflows on separable metric spaces, Far East Journal of Dynamical Systems, 17 (2011), 49-54.  Google Scholar

K. P. Persidskii, On a theorem of Liapunov, C. R. (Dokl.) Acad. Sci. URSS, 14 (1937), 541-543. Google Scholar

V. M. Popov, Absolute stability of nonlinear systems of automatic control, Automation and Remote Control, 22 (1961), 857-875.  Google Scholar

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. Google Scholar

V. M. Popov, Hyperstability of Control Systems, Springer-Verlag, 1973, Translated from the Romanian Edition of 1966.  Google Scholar

A. Rantzer, A dual to Lyapunov's stability theorem, Systems & Control Letters, 42 (2001), 161-168. doi: 10.1016/S0167-6911(00)00087-6.  Google Scholar

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.  Google Scholar

L. Rifford, Existence of Lipschitz and semiconcave control-Lyapunov functions, SIAM Journal on Control and Optimization, 39 (2000), 1043-1064. doi: 10.1137/S0363012999356039.  Google Scholar

L. Rifford, Semiconcave control-Lyapunov functions and stabilizing feedbacks, SIAM Journal on Control and Optimization, 41 (2002), 659-681. doi: 10.1137/S0363012900375342.  Google Scholar

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.  Google Scholar

N. Rouche, P. Habets and M. Laloy, Stability Theory by Liapunov's Direct Method, Springer-Verlag, 1977.  Google Scholar

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.  Google Scholar

E. Roxin, Stability in general control systems, Journal of Differential Equations, 1 (1965), 115-150. doi: 10.1016/0022-0396(65)90015-X.  Google Scholar

E. Roxin, On asymptotic stability in control systems, Rendiconti del Circolo Matematico di Palermo, 15 (1966), 193-208. doi: 10.1007/BF02849435.  Google Scholar

E. Roxin, On stability in control systems, SIAM Journal on Control, 3 (1966), 357-372. doi: 10.1137/0303024.  Google Scholar

C. E. Shannon and W. Weaver, The Mathematical Theory of Communication, University of Illinois Press, 1949.  Google Scholar

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.  Google Scholar

D. D. Šiljak, Nonlinear Systems: The Parameter Analysis and Design, John Wiley & Sons Inc., 1969. Google Scholar

G. V. Smirnov, Weak asymptotic stability of differential inclusions I, Automation and Remote Control, 51 (1990), 901-908.  Google Scholar

G. V. Smirnov, Weak asymptotic stability of differential inclusions II, Automation and Remote Control, 51 (1990), 1052-1058.  Google Scholar

E. D. Sontag, A Lyapunov-like characterization of asymptotic controllability, SIAM Journal on Control and Optimization, 21 (1983), 462-471. doi: 10.1137/0321028.  Google Scholar

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34 (1989), 435-443. doi: 10.1109/9.28018.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

Y. Z. Tsypkin, The absolute stability of large-scale nonlinear sampled-data systems, (Russian) Doklady Akademii Nauk SSSR, 145 (1962), 52-55. Google Scholar

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.  Google Scholar

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.  Google Scholar

I. Vrkoč, A general theorem of Chetaev, (Russian) Czechoslovak Mathematical Journal, 5 (1955), 451-461. Google Scholar

J. C. Willems, Dissipative dynamical systems part I: General theory, Archive for Rational Mechanics and Analysis, 45 (1972), 321-351. doi: 10.1007/BF00276493.  Google Scholar

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.  Google Scholar

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.  Google Scholar

V. A. Yakubovich, The solution of certain matrix inequalities in automatic control theory, Doklady Akademii Nauk SSSR, 143 (1962), 1304-1307.  Google Scholar

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.  Google Scholar

T. Yoshizawa, Stability Theory by Liapunov's Second Method, Mathematical Society of Japan, 1966.  Google Scholar

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.  Google Scholar