2015, 20(8): 2333-2360. doi: 10.3934/dcdsb.2015.20.2333

Classical converse theorems in Lyapunov's second method

1. 

School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, New South Wales 2308

Received  August 2014 Revised  March 2015 Published  August 2015

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.
Citation: Christopher M. Kellett. Classical converse theorems in Lyapunov's second method. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2333-2360. doi: 10.3934/dcdsb.2015.20.2333
References:
[1]

B. D. O. Anderson, Stability of control systems with multiple nonlinearities,, Journal of the Franklin Institute, 282 (1966), 155. 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. 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. 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. doi: 10.1016/S0167-6911(99)00055-9.

[5]

H. Antosiewicz, A survey of Lyapunov's second method,, Contributions to Nonlinear Oscillations, (1958), 147.

[6]

T. M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus,, Addison-Wesley Publishing Company, (1957).

[7]

L. Arnold and B. Schmalfuss, Lyapunov's second method for random dynamical systems,, Journal of Differential Equations, 177 (2001), 235. doi: 10.1006/jdeq.2000.3991.

[8]

A. Bacciotti and L. Rosier, Liapunov and Lagrange stability: Inverse theorems for discontinuous systems,, Mathematics of Control, 11 (1998), 101. doi: 10.1007/BF02741887.

[9]

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

[10]

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

[11]

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

[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.

[13]

R. W. Brockett, Asymptotic stability and feedback stabilization,, in Differential Geometric Control Theory (eds. R. W. Brockett, (1983), 181.

[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. 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. doi: 10.1109/TAC.2008.919257.

[16]

N. G. Chetayev, The Stability of Motion,, Pergamon Press, (1961).

[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. 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. 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. 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, (1998).

[21]

C. Conley, Isolated Invariant Sets and the Morse Index,, CBMS Regional Conference Series no. 38, (1978).

[22]

T. M. Cover and J. A. Thomas, Elements of Information Theory,, 2nd edition, (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, 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. 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. 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. 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. doi: 10.1137/100787829.

[31]

W. Hahn, Theory and Application of Liapunov's Direct Method,, Prentice-Hall, (1963).

[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. 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. 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, (2002), 3946. 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. 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. 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, 82 (1960), 371. 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, 82 (1960), 394. 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. 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. doi: 10.1137/S0363012901392967.

[42]

C. M. Kellett, A compendium of comparison function results,, Mathematics of Controls, 26 (2014), 339. 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, (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, (2000), 3994. 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. 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. 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. 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, 19 (2007), 183. doi: 10.1007/s00498-007-0016-6.

[49]

H. K. Khalil, Nonlinear Systems,, 2nd edition, (1996).

[50]

R. Khasminskii, Stochastic Stability of Differential Equations,, 2nd edition, (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), (1977), 119.

[52]

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

[53]

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

[54]

P. Kokotović and M. Arcak, Constructive nonlinear control: A historical perspective,, Automatica, 37 (2001), 637. 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.

[56]

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

[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.

[58]

N. N. Krasovskii, Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay,, Stanford University Press, (1963).

[59]

M. Krichman, E. D. Sontag and Y. Wang, Input-output-to-state stability,, SIAM Journal on Control and Optimization, 39 (2001), 1874. doi: 10.1137/S0363012999365352.

[60]

M. Krstić, I. Kanellakopoulos and P. Kokotović, Nonlinear and Adaptive Control Design,, John Wiley and Sons, (1995).

[61]

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

[62]

J. Kurzweil, On the inversion of Ljapunov's second theorem on stability of motion,, (Russian) Czechoslovak Mathematical Journal, 81 (1956), 217.

[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.

[64]

H. J. Kushner, Converse theorems for stochastic Liapunov functions,, SIAM Journal on Control and Optimization, 5 (1967), 228. 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.

[67]

A. L. Letov, Stability in Nonlinear Control Systems,, Princeton University Press, (1961).

[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. 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).

[70]

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

[71]

A. M. Lyapunov, The general problem of the stability of motion,, (Russian) Math. Soc. of Kharkov; English Translation, 55 (1992), 531. 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.

[73]

I. G. Malkin, Some Problems in the Theory of Nonlinear Oscillations,, United States Atomic Energy Commission, (1959).

[74]

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

[75]

J. L. Massera, Contributions to stability theory,, Annals of Mathematics, 64 (1956), 182. doi: 10.2307/1969955.

[76]

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

[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.

[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.

[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.

[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. 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. 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.

[84]

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

[85]

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

[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, 14 (1964), 913.

[87]

V. M. Popov, Hyperstability of Control Systems,, Springer-Verlag, (1973).

[88]

A. Rantzer, A dual to Lyapunov's stability theorem,, Systems & Control Letters, 42 (2001), 161. 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, (2002), 1890. 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. doi: 10.1137/S0363012999356039.

[91]

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

[92]

L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field,, Systems & Control Letters, 19 (1992), 467. 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. doi: 10.1016/0022-0396(65)90019-7.

[95]

E. Roxin, Stability in general control systems,, Journal of Differential Equations, 1 (1965), 115. 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. doi: 10.1007/BF02849435.

[97]

E. Roxin, On stability in control systems,, SIAM Journal on Control, 3 (1966), 357. 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. 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.

[102]

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

[103]

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

[104]

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

[105]

E. D. Sontag, Clocks and insensitivity to small measurement errors,, ESAIM: Control, 4 (1999), 537. 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. 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. 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. 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. 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, 5 (2000), 313. 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.

[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.

[113]

V. I. Vorotnikov, Partial stability and control: The state-of-the-art and development prospects,, Automation and Remote Control, 66 (2005), 511. doi: 10.1007/s10513-005-0099-9.

[114]

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

[115]

J. C. Willems, Dissipative dynamical systems part I: General theory,, Archive for Rational Mechanics and Analysis, 45 (1972), 321. 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. doi: 10.1007/BF00276494.

[117]

F. W. Wilson, Smoothing derivatives of functions and applications,, Transactions of the American Mathematical Society, 139 (1969), 413. 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.

[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.

[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, (1964).

show all references

References:
[1]

B. D. O. Anderson, Stability of control systems with multiple nonlinearities,, Journal of the Franklin Institute, 282 (1966), 155. 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. 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. 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. doi: 10.1016/S0167-6911(99)00055-9.

[5]

H. Antosiewicz, A survey of Lyapunov's second method,, Contributions to Nonlinear Oscillations, (1958), 147.

[6]

T. M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus,, Addison-Wesley Publishing Company, (1957).

[7]

L. Arnold and B. Schmalfuss, Lyapunov's second method for random dynamical systems,, Journal of Differential Equations, 177 (2001), 235. doi: 10.1006/jdeq.2000.3991.

[8]

A. Bacciotti and L. Rosier, Liapunov and Lagrange stability: Inverse theorems for discontinuous systems,, Mathematics of Control, 11 (1998), 101. doi: 10.1007/BF02741887.

[9]

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

[10]

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

[11]

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

[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.

[13]

R. W. Brockett, Asymptotic stability and feedback stabilization,, in Differential Geometric Control Theory (eds. R. W. Brockett, (1983), 181.

[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. 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. doi: 10.1109/TAC.2008.919257.

[16]

N. G. Chetayev, The Stability of Motion,, Pergamon Press, (1961).

[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. 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. 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. 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, (1998).

[21]

C. Conley, Isolated Invariant Sets and the Morse Index,, CBMS Regional Conference Series no. 38, (1978).

[22]

T. M. Cover and J. A. Thomas, Elements of Information Theory,, 2nd edition, (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, 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. 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. 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. 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. doi: 10.1137/100787829.

[31]

W. Hahn, Theory and Application of Liapunov's Direct Method,, Prentice-Hall, (1963).

[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. 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. 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, (2002), 3946. 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. 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. 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, 82 (1960), 371. 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, 82 (1960), 394. 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. 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. doi: 10.1137/S0363012901392967.

[42]

C. M. Kellett, A compendium of comparison function results,, Mathematics of Controls, 26 (2014), 339. 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, (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, (2000), 3994. 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. 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. 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. 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, 19 (2007), 183. doi: 10.1007/s00498-007-0016-6.

[49]

H. K. Khalil, Nonlinear Systems,, 2nd edition, (1996).

[50]

R. Khasminskii, Stochastic Stability of Differential Equations,, 2nd edition, (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), (1977), 119.

[52]

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

[53]

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

[54]

P. Kokotović and M. Arcak, Constructive nonlinear control: A historical perspective,, Automatica, 37 (2001), 637. 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.

[56]

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

[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.

[58]

N. N. Krasovskii, Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay,, Stanford University Press, (1963).

[59]

M. Krichman, E. D. Sontag and Y. Wang, Input-output-to-state stability,, SIAM Journal on Control and Optimization, 39 (2001), 1874. doi: 10.1137/S0363012999365352.

[60]

M. Krstić, I. Kanellakopoulos and P. Kokotović, Nonlinear and Adaptive Control Design,, John Wiley and Sons, (1995).

[61]

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

[62]

J. Kurzweil, On the inversion of Ljapunov's second theorem on stability of motion,, (Russian) Czechoslovak Mathematical Journal, 81 (1956), 217.

[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.

[64]

H. J. Kushner, Converse theorems for stochastic Liapunov functions,, SIAM Journal on Control and Optimization, 5 (1967), 228. 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.

[67]

A. L. Letov, Stability in Nonlinear Control Systems,, Princeton University Press, (1961).

[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. 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).

[70]

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

[71]

A. M. Lyapunov, The general problem of the stability of motion,, (Russian) Math. Soc. of Kharkov; English Translation, 55 (1992), 531. 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.

[73]

I. G. Malkin, Some Problems in the Theory of Nonlinear Oscillations,, United States Atomic Energy Commission, (1959).

[74]

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

[75]

J. L. Massera, Contributions to stability theory,, Annals of Mathematics, 64 (1956), 182. doi: 10.2307/1969955.

[76]

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

[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.

[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.

[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.

[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. 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. 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.

[84]

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

[85]

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

[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, 14 (1964), 913.

[87]

V. M. Popov, Hyperstability of Control Systems,, Springer-Verlag, (1973).

[88]

A. Rantzer, A dual to Lyapunov's stability theorem,, Systems & Control Letters, 42 (2001), 161. 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, (2002), 1890. 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. doi: 10.1137/S0363012999356039.

[91]

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

[92]

L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field,, Systems & Control Letters, 19 (1992), 467. 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. doi: 10.1016/0022-0396(65)90019-7.

[95]

E. Roxin, Stability in general control systems,, Journal of Differential Equations, 1 (1965), 115. 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. doi: 10.1007/BF02849435.

[97]

E. Roxin, On stability in control systems,, SIAM Journal on Control, 3 (1966), 357. 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. 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.

[102]

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

[103]

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

[104]

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

[105]

E. D. Sontag, Clocks and insensitivity to small measurement errors,, ESAIM: Control, 4 (1999), 537. 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. 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. 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. 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. 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, 5 (2000), 313. 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.

[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.

[113]

V. I. Vorotnikov, Partial stability and control: The state-of-the-art and development prospects,, Automation and Remote Control, 66 (2005), 511. doi: 10.1007/s10513-005-0099-9.

[114]

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

[115]

J. C. Willems, Dissipative dynamical systems part I: General theory,, Archive for Rational Mechanics and Analysis, 45 (1972), 321. 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. doi: 10.1007/BF00276494.

[117]

F. W. Wilson, Smoothing derivatives of functions and applications,, Transactions of the American Mathematical Society, 139 (1969), 413. 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.

[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.

[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, (1964).

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