# American Institute of Mathematical Sciences

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:
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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. 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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. 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