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Review on computational methods for Lyapunov functions

Abstract / Introduction Related Papers Cited by
  • Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them.
        Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ different methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function.
    Mathematics Subject Classification: Primary: 37M99, 34D20; Secondary: 34D05, 37C75, 34D45.

    Citation:

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  • [1]

    N. Aghannan and P. Rouchon, An intrinsic observer for a class of Lagrangian systems, IEEE Trans. Automat. Control, 48 (2003), 936-945.doi: 10.1109/TAC.2003.812778.

    [2]

    A. Agrachev and D. Liberzon, Lie-algebraic stability criteria for switched systems, SIAM J. Control Optim., 40 (2001), 253-269.doi: 10.1137/S0363012999365704.

    [3]

    A. Ahmadi and R. Jungers, On complexity of Lyapunov functions for switched linear systems, in Proceedings of the 19th World Congress of the International Federation of Automatic Control, Cape Town, South Africa, 2014.

    [4]

    A. Ahmadi, K. Krstic and P. Parrilo, A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function, in Proceedings of the 50th IEEE Conference on Decision and Control (CDC), Orlando (FL), USA, 2011, 7579-7580.doi: 10.1109/CDC.2011.6161499.

    [5]

    A. Ahmadi, A. Majumdar and R. Tedrake, Complexity of ten decision problems in continuous time dynamical systems, in Proceedings of the American Control Conference, Washington (DC), USA, 2013, 6376-6381.doi: 10.1109/ACC.2013.6580838.

    [6]

    E. Akin, The General Topology of Dynamical Systems, American Mathematical Society, 1993.

    [7]

    A. Aleksandrov, A. Martynyuk and A. Zhabko, Professor V. I. Zubov to the 80th birthday anniversary, Nonlinear Dyn. Syst. Theory, 10 (2010), 1-10.

    [8]

    R. Ambrosino and E. Garone, Robust stability of linear uncertain systems through piecewise quadratic Lyapunov functions defined over conical partitions, in Proceedings of the 51st IEEE Conference on Decision and Control, Maui (HI), USA, 2012, 2872-2877.doi: 10.1109/CDC.2012.6427016.

    [9]

    J. Anderson and A. Papachristodoulou, Advances in computational Lyapunov analysis using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 8 (2015), 2361-2381.doi: 10.3934/dcdsb.2015.20.2361.

    [10]

    D. Angeli, A Lyapunov approach to incremental stability properties, IEEE Trans. Automat. Contr., 47 (2002), 410-421.doi: 10.1109/9.989067.

    [11]

    E. Aragão-Costa, T. Caraballo, A. Carvalho and J. Langa, Stability of gradient semigroups under perturbations, Nonlinearity, 24 (2011), 2099-2117.doi: 10.1088/0951-7715/24/7/010.

    [12]

    E. Aragão-Costa, T. Caraballo, A. Carvalho and J. Langa, Non-autonomous Morse-decomposition and Lyapunov functions for gradient-like processes, Trans. Amer. Math. Soc., 365 (2013), 5277-5312.doi: 10.1090/S0002-9947-2013-05810-2.

    [13]

    L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, 1974.

    [14]

    L. Arnold, Random dynamical systems, in Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, 1995, 1-43.doi: 10.1007/BFb0095238.

    [15]

    L. Arnold and B. Schmalfuss, Lyapunov's second method for random dynamical systems, J. Differential Equations, 177 (2001), 235-265.doi: 10.1006/jdeq.2000.3991.

    [16]

    J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, 1984.doi: 10.1007/978-3-642-69512-4.

    [17]

    B. Aulbach, Asymptotic stability regions via extensions of Zubov's method. I, II, Nonlinear Anal., 7 (1983), 1431-1440, 1441-1454.doi: 10.1016/0362-546X(83)90010-X.

    [18]

    E. Aylward, P. Parrilo and J.-J. Slotine, Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming, Automatica, 44 (2008), 2163-2170.doi: 10.1016/j.automatica.2007.12.012.

    [19]

    R. Baier, L. Grüne and S. Hafstein, Linear programming based Lyapunov function computation for differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 33-56.doi: 10.3934/dcdsb.2012.17.33.

    [20]

    R. Baier and S. Hafstein, Numerical computation of Control Lyapunov Functions in the sense of generalized gradients, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS) (no. 0232), Groningen, The Netherlands, 2014, 1173-1180.

    [21]

    H. Ban and W. Kalies, A computational approach to Conley's decomposition theorem, J. Comput. Nonlinear Dynam., 1 (2006), 312-319.doi: 10.1115/1.2338651.

    [22]

    E. Barbašin and N. Krasovskiĭ, On the existence of Lyapunov functions in the case of asymptotic stability in the large, Prikl. Mat. Meh., 18 (1954), 345-350.

    [23]

    R. Bartels and G. Stewart, Solution of the matrix equation AX+XB=C, Communications of the ACM, 15 (1972), 820-826.doi: 10.1145/361573.361582.

    [24]

    R. Bellman, Vector Lyapunov functions, J. SIAM Control Ser. A, 1 (1962), 32-34.

    [25]

    R. Bellman, Introduction to Matrix Analysis, Classics in Applied Mathematics, 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia (PA), USA, 1995.

    [26]

    A. Berger, On finite-time hyperbolicity, Commun. Pure Appl. Anal., 10 (2011), 963-981.doi: 10.3934/cpaa.2011.10.963.

    [27]

    A. Berger, T. S. Doan and S. Siegmund, A definition of spectrum for differential equations on finite time, J. Differential Equations, 246 (2009), 1098-1118.doi: 10.1016/j.jde.2008.06.036.

    [28]

    J. Bernussou and P. Peres, A linear programming oriented procedure for quadratic stabilization of uncertain systems, Systems Control Lett., 13 (1989), 65-72.doi: 10.1016/0167-6911(89)90022-4.

    [29]

    N. Bhatia and G. Szegő, Dynamical Systems: Stability Theory and Applications, Lecture Notes in Mathematics, 35, Springer, Berlin, 1967.

    [30]

    G. Birkhoff, Dynamical Systems, American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, R.I., 1966.

    [31]

    J. Björnsson, P. Giesl and S. Hafstein, Algorithmic verification of approximations to complete Lyapunov functions, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (no. 0180), Groningen, The Netherlands, 2014, 1181-1188.

    [32]

    J. Björnsson, P. Giesl, S. Hafstein, C. Kellett and H. Li, Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction, in Proceedings of the CDC, 53rd IEEE Conference on Decision and Control, Los Angeles, California, USA, 2014, 5506-5511.

    [33]

    J. Björnsson, P. Giesl, S. Hafstein, C. Kellett and H. Li, Computation of Lyapunov functions for systems with multiple attractors, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 4019-4039.doi: 10.3934/dcds.2015.35.4019.

    [34]

    F. Blanchini, Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions, in Proceedings of the 30th IEEE Conference on Decision and Control, Vol. 2, 1991, 1755-1760.doi: 10.1109/CDC.1991.261708.

    [35]

    F. Blanchini, Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions, IEEE Trans. Automat. Control, 39 (1994), 428-433.doi: 10.1109/9.272351.

    [36]

    F. Blanchini, Nonquadratic Lyapunov functions for robust control, Automatica, 31 (1995), 451-461.doi: 10.1016/0005-1098(94)00133-4.

    [37]

    F. Blanchini and S. Carabelli, Robust stabilization via computer-generated Lyapunov functions: An application to a magnetic levitation system, in Proceedings of the 33th IEEE Conference on Decision and Control, Vol. 2, 1994, 1105-1106.doi: 10.1109/CDC.1994.411291.

    [38]

    F. Blanchini and S. Miani, Set-theoretic Methods in Control, Systems & Control: Foundations & Applications, Birkhäuser, 2008.

    [39]

    V. Boichenko, G. Leonov and V. Reitmann, Dimension Theory for Ordinary Differential Equations, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 141, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 2005.doi: 10.1007/978-3-322-80055-8.

    [40]

    G. Borg, A Condition for the Existence Of Orbitally Stable Solutions of Dynamical Systems, Kungliga Tekniska Högskolan Handlingar Stockholm, 153, 1960.

    [41]

    J. Bouvrie and B. Hamzi, Model reduction for nonlinear control systems using kernel subspace methods, arXiv:1108.2903v2, 2011.

    [42]

    S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, 15, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994.doi: 10.1137/1.9781611970777.

    [43]

    S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.doi: 10.1017/CBO9780511804441.

    [44]

    M. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Control, 43 (1998), 475-482.doi: 10.1109/9.664150.

    [45]

    R. Brayton and C. Tong, Stability of dynamical systems: A constructive approach, IEEE Trans. Circuits and Systems, 26 (1979), 224-234.doi: 10.1109/TCS.1979.1084637.

    [46]

    R. Brayton and C. Tong, Constructive stability and asymptotic stability of dynamical systems, IEEE Trans. Circuits and Systems, 27 (1980), 1121-1130.doi: 10.1109/TCS.1980.1084749.

    [47]

    M. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics, 12, Cambridge University Press, Cambridge, 2003.doi: 10.1017/CBO9780511543241.

    [48]

    H. Burchardt and S. Ratschan, Estimating the region of attraction of ordinary differential equations by quantified constraint solving, in Proceedings Of The 3rd WSEAS International Conference On Dynamical Systems And Control, 2007, 241-246.

    [49]

    C. Byrnes, Topological methods for nonlinear oscillations, Notices Amer. Math. Soc., 57 (2010), 1080-1091.

    [50]

    F. Camilli, L. Grüne and F. Wirth, A generalization of Zubov's method to perturbed systems, SIAM J. Control Optim., 40 (2001), 496-515.doi: 10.1137/S036301299936316X.

    [51]

    F. Camilli, L. Grüne and F. Wirth, A regularization of Zubov's equation for robust domains of attraction, in Nonlinear Control in the Year 2000, Vol. 1 (Paris), Lecture Notes in Control and Inform. Sci., 258, Springer, London, 2001, 277-289.doi: 10.1007/BFb0110220.

    [52]

    F. Camilli, L. Grüne and F. Wirth, Control Lyapunov functions and Zubov's method, SIAM J. Control Optim., 47 (2008), 301-326.doi: 10.1137/06065129X.

    [53]

    A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013.doi: 10.1007/978-1-4614-4581-4.

    [54]

    C. Chen and E. Kinnen, Construction of Liapunov functions, J. Franklin Inst., 289 (1970), 133-146.doi: 10.1016/0016-0032(70)90299-1.

    [55]

    G. Chesi, LMI techniques for optimization over polynomials in control: A survey, IEEE Trans. Automat. Control, 55 (2010), 2500-2510.doi: 10.1109/TAC.2010.2046926.

    [56]

    C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 1999.

    [57]

    I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, ACTA Scientific Publishing House, Kharkiv, Ukraine, 2002.

    [58]

    F. Clarke, Lyapunov functions and discontinuous stabilizing feedback, Annu. Rev. Control, 35 (2011), 13-33.doi: 10.1016/j.arcontrol.2011.03.001.

    [59]

    F. Clarke, Y. Ledyaev and R. Stern, Asymptotic stability and smooth Lyapunov functions, J. Differential Equations, 149 (1998), 69-114.doi: 10.1006/jdeq.1998.3476.

    [60]

    C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series, 38, American Mathematical Society, 1978.

    [61]

    J.-M. Coron, B. d'Andréa Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, 52 (2007), 2-11.doi: 10.1109/TAC.2006.887903.

    [62]

    E. Davison and E. Kurak, A computational method for determining quadratic Lyapunov functions for non-linear systems, Automatica, 7 (1971), 627-636.doi: 10.1016/0005-1098(71)90027-6.

    [63]

    G. Davrazos and N. Koussoulas, A review of stability results for switched and hybrid systems, in Proceedings of 9th Mediterranean Conference on Control and Automation, Dubrovnik, Croatia, 2001.

    [64]

    W. Dayawansa and C. Martin, A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE Tra, 44 (1999), 751-760.doi: 10.1109/9.754812.

    [65]

    M. Dellnitz, G. Froyland and O. Junge, The algorithms behind {GAIO} - set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001, 145-174, 805-807.

    [66]

    M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002, 221-264.doi: 10.1016/S1874-575X(02)80026-1.

    [67]

    U. Dini, Fondamenti per la Teoria Delle Funzioni di Variabili Reali, (in Italian) Pisa, 1878.

    [68]

    S. Dubljević and N. Kazantzis, A new Lyapunov design approach for nonlinear systems based on Zubov's method, Automatica, 38 (2002), 1999-2007.doi: 10.1016/S0005-1098(02)00110-3.

    [69]

    N. Eghbal, N. Pariz and A. Karimpour, Discontinuous piecewise quadratic Lyapunov functions for planar piecewise affine systems, J. Math. Anal. Appl., 399 (2013), 586-593.doi: 10.1016/j.jmaa.2012.09.054.

    [70]

    K. Erickson and A. Michel, Stability analysis of fixed-point digital filters using computer generated Lyapunov functions - Part I: Direct form and coupled form filtes, IEEE Trans. Circuits and Systems, 32 (1985), 113-132.doi: 10.1109/TCS.1985.1085676.

    [71]

    K. Erickson and A. Michel, Stability analysis of fixed-point digital filters using computer generated Lyapunov functions - Part II: Wave digital filters and lattice digital filters, IEEE Trans. Circuits and Systems, 32 (1985), 132-142.doi: 10.1109/TCS.1985.1085677.

    [72]

    M. Falcone, Numerical solution of dynamic programming equations, in Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Appendix A, Birkhäuser, Boston, 1997.

    [73]

    F. Fallside, M. Patel, M. Etherton, S. Margolis and W. Vogt, Control engineering applications of V. I. Zubov's construction procedure for Lyapunov functions, IEEE Trans. Automat. Control, 10 (1965), 220-222.doi: 10.1109/TAC.1965.1098103.

    [74]

    F. Faria, G. Silva and V. Oliveira, Reducing the conservatism of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy Lyapunov functions, International Journal of Systems Science, 44 (2013), 1956-1969.doi: 10.1080/00207721.2012.670307.

    [75]

    D. R. Ferguson, Generalisation of Zubov's construction procedure for Lyapunov functions, Electron. Lett., 6 (1970), 73-74.doi: 10.1049/el:19700046.

    [76]

    A. Filippov, Differential Equations with Discontinuous Right-hand Side, Translated from Russian, original book from 1985, Kluwer, 1988.

    [77]

    H. Flashner and R. Guttalu, A computational approach for studying domains of attraction for nonlinear systems, Internat. J. Non-Linear Mech., 23 (1988), 279-295.doi: 10.1016/0020-7462(88)90026-1.

    [78]

    F. Forni and R. Sepulchre, A differential Lyapunov framework for Contraction Analysis, IEEE Trans. Automat. Control, 59 (2014), 614-628.doi: 10.1109/TAC.2013.2285771.

    [79]

    K. Forsman, Construction of Lyapunov functions using Grobner bases, In Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, UK, 1 (1991), 798-799.doi: 10.1109/CDC.1991.261424.

    [80]

    R. Geiselhart, R. Gielen, M. Lazar and F. Wirth, An alternative converse Lyapunov theorem for discrete-time systems, Systems Control Lett., 70 (2014), 49-59.doi: 10.1016/j.sysconle.2014.05.007.

    [81]

    R. Genesio, M. Tartaglia and A. Vicino, On the estimation of asymptotic stability regions: State of the art and new proposals, IEEE Trans. Automat. Control, 30 (1985), 747-755.doi: 10.1109/TAC.1985.1104057.

    [82]

    P. Giesl, Necessary conditions for a limit cycle and its basin of attraction, Nonlinear Anal., 56 (2004), 643-677.doi: 10.1016/j.na.2003.07.020.

    [83]

    P. Giesl, The basin of attraction of periodic orbits in nonsmooth differential equations, ZAMM Z. Angew. Math. Mech., 85 (2005), 89-104.doi: 10.1002/zamm.200310164.

    [84]

    P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math., 1904, Springer, 2007.

    [85]

    P. Giesl, On the determination of the basin of attraction of discrete dynamical systems, J. Difference Equ. Appl., 13 (2007), 523-546.doi: 10.1080/10236190601135209.

    [86]

    P. Giesl, Construction of a local and global Lyapunov function using radial basis functions, IMA J. Appl. Math., 73 (2008), 782-802.doi: 10.1093/imamat/hxn018.

    [87]

    P. Giesl, On the determination of the basin of attraction of periodic orbits in three- and higher-dimensional systems, J. Math. Anal. Appl., 354 (2009), 606-618.doi: 10.1016/j.jmaa.2009.01.027.

    [88]

    P. Giesl, Construction of a finite-time Lyapunov function by meshless collocation, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2387-2412.doi: 10.3934/dcdsb.2012.17.2387.

    [89]

    P. Giesl, Converse theorems on contraction metrics for an equilibrium, J. Math. Anal. Appl., 424 (2015), 1380-1403.doi: 10.1016/j.jmaa.2014.12.010.

    [90]

    P. Giesl and S. Hafstein, Existence of piecewise affine Lyapunov functions in two dimensions, J. Math. Anal. Appl., 371 (2010), 233-248.doi: 10.1016/j.jmaa.2010.05.009.

    [91]

    P. Giesl and S. Hafstein, Existence of piecewise linear Lyapunov functions in arbitrary dimensions, Discrete Contin. Dyn. Syst., 32 (2012), 3539-3565.doi: 10.3934/dcds.2012.32.3539.

    [92]

    P. Giesl and S. Hafstein, Construction of a CPA contraction metric for periodic orbits using semidefinite optimization, Nonlinear Anal., 86 (2013), 114-134.doi: 10.1016/j.na.2013.03.012.

    [93]

    P. Giesl and S. Hafstein, Computation of Lyapunov functions for nonlinear discrete time systems by linear programming, J. Difference Equ. Appl., 20 (2014), 610-640.doi: 10.1080/10236198.2013.867341.

    [94]

    P. Giesl and S. Hafstein, Revised CPA method to compute Lyapunov functions for nonlinear systems, J. Math. Anal. Appl., 410 (2014), 292-306.doi: 10.1016/j.jmaa.2013.08.014.

    [95]

    P. Giesl and M. Rasmussen, Areas of attraction for nonautonomous differential equations on finite time intervals, J. Math. Anal. Appl., 390 (2012), 27-46.doi: 10.1016/j.jmaa.2011.12.051.

    [96]

    P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Numer. Anal., 45 (2007), 1723-1741.doi: 10.1137/060658813.

    [97]

    P. Giesl and H. Wendland, Approximating the basin of attraction of time-periodic {ODE}s by meshless collocation, Discrete Contin. Dyn. Syst., 25 (2009), 1249-1274.doi: 10.3934/dcds.2009.25.1249.

    [98]

    P. Giesl and H. Wendland, Numerical determination of the basin of attraction for asymptotically autonomous dynamical systems, Nonlinear Anal., 75 (2012), 2823-2840.doi: 10.1016/j.na.2011.11.027.

    [99]

    R. Goebel, R. Sanfelice and A. Teel, Hybrid Dynamical Systems, Modeling, stability, and robustness, Princeton University Press, Princeton, NJ, 2012.

    [100]

    R. Goebel, A. Teel, T. Hu and Z. Lin, Conjugate convex Lyapunov functions for dual linear differential inclusions, IEEE Trans. Automat. Control, 51 (2006), 661-666.doi: 10.1109/TAC.2006.872764.

    [101]

    A. Goullet, S. Harker, K. Mischaikow, W. Kalies and D. Kasti, Efficient computation of Lyapunov functions for Morse decompositions, Discrete Contin. Dyn. Syst. Ser. B, 8 (2015), 2419-2451.doi: 10.3934/dcdsb.2015.20.2419.

    [102]

    B. Grosman and D. Lewin, Lyapunov-based stability analysis automated by genetic programming, Automatica, 45 (2009), 252-256.doi: 10.1016/j.automatica.2008.07.014.

    [103]

    L. Grujić, Exact determination of a Lyapunov function and the asymptotic stability domain, Internat. J. Systems Sci., 23 (1992), 1871-1888.doi: 10.1080/00207729208949427.

    [104]

    L. Grujić, Complete exact solution to the Lyapunov stability problem: Time-varying nonlinear systems with differentiable motions, Nonlinear Anal., 22 (1994), 971-981.doi: 10.1016/0362-546X(94)90060-4.

    [105]

    L. Grüne, Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization, Lecture Notes in Mathematics, 1783, Springer-Verlag, Berlin, 2002.doi: 10.1007/b83677.

    [106]

    L. Grüne, P. Kloeden, S. Siegmund and F. Wirth, Lyapunov's second method for nonautonomous differential equations, Discrete Contin. Dyn. Syst., 18 (2007), 375-403.doi: 10.3934/dcds.2007.18.375.

    [107]

    O. $\ddot G$urel and L. Lapidus, A guide to the generation of Liapunov functions, Indust. Engrg. Chem., 61 (1969), 30-41.doi: 10.1021/ie50711a006.

    [108]

    S. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete Contin. Dyn. Syst., 10 (2004), 657-678.doi: 10.3934/dcds.2004.10.657.

    [109]

    S. Hafstein, A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations, Dynamical Systems: An International Journal, 20 (2005), 281-299.doi: 10.1080/14689360500164873.

    [110]

    S. Hafstein, An Algorithm for Constructing Lyapunov Functions, Electronic Journal of Differential Equations. Monograph, 8, Texas State University-San Marcos, Department of Mathematics, San Marcos, TX, 2007. Available from: http://ejde.math.txstate.edu/.

    [111]

    S. Hafstein, C. Kellett and H. Li, Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction, in Proceedings of the 2014 American Control Conference (no. 0170), Portland (OR), USA, 2014, 548-553.doi: 10.1109/ACC.2014.6858660.

    [112]

    W. Hahn, Stability of Motion, Springer, Berlin, 1967.

    [113]

    G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10 (2000), 99-108.doi: 10.1063/1.166479.

    [114]

    B. Hargrave, Using Genetic Algorithms to Optimize Control Lyapunov Functions, PhD thesis, Oklahoma State University, 2008.

    [115]

    P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.

    [116]

    P. Hartman and C. Olech, On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc., 104 (1962), 154-178.

    [117]

    M. Abu Hassan and C. Storey, Numerical determination of domains of attraction for electrical power systems using the method of Zubov, Int. J. Control, 34 (1981), 371-381.doi: 10.1080/00207178108922536.

    [118]

    C. Hsu, Cell-to-cell Mapping, Applied Mathematical Sciences, 64, Springer-Verlag, New York, 1987.doi: 10.1007/978-1-4757-3892-6.

    [119]

    T. Hu and Z. Lin, Composite quadratic Lyapunov functions for constrained control systems, IEEE Trans. Automat. Control, 48 (2003), 440-450.doi: 10.1109/TAC.2003.809149.

    [120]

    M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. Amer. Math. Soc., 126 (1998), 245-256.doi: 10.1090/S0002-9939-98-04500-6.

    [121]

    B. Ingalls, E. Sontag and Y. Wang, An infinite-time relaxation theorem for differential inclusions, Proc. Amer. Math. Soc., 131 (2003), 487-499.doi: 10.1090/S0002-9939-02-06539-5.

    [122]

    T. Johansen, Computation of Lyapunov functions for smooth, nonlinear systems using convex optimization, Automatica, 36 (2000), 1617-1626.doi: 10.1016/S0005-1098(00)00088-1.

    [123]

    M. Johansson, Piecewise Linear Control Systems, Lecture Notes in Control and Information Sciences, 2003.doi: 10.1007/3-540-36801-9.

    [124]

    M. Johansson and A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Trans. Automat. Control, 43 (1998), 555-559.doi: 10.1109/9.664157.

    [125]

    P. Julian, A High Level Canonical Piecewise Linear Representation: Theory and Applications, PhD thesis, Universidad Nacional del Sur, Bahia Blanca, Argentina, 1999.

    [126]

    P. Julian, J. Guivant and A. Desages, A parametrization of piecewise linear Lyapunov functions via linear programming, Int. J. Control, 72 (1999), 702-715.doi: 10.1080/002071799220876.

    [127]

    O. Junge, Rigorous discretization of subdivision techniques, in International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 916-918.

    [128]

    W. Kalies, K. Mischaikow and R. VanderVorst, An algorithmic approach to chain recurrence, Found. Comput. Math, 5 (2005), 409-449.doi: 10.1007/s10208-004-0163-9.

    [129]

    R. Kamyar and M. Peet, Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares, Discrete Contin. Dyn. Syst. Ser. B, 2383-2417. doi: 10.3934/dcdsb.2015.20.2383.

    [130]

    J. Kapinski, J. Deshmukh, S. Sankaranarayanan and N. Arechiga, Simulation-guided Lyapunov analysis for hybrid dynamical systems, in Proceedings of the 17th International Conference on Hybrid Systems: Computation and Control (HSCC 2014), Berlin, Germany, 2014, 133-142.doi: 10.1145/2562059.2562139.

    [131]

    C. Kellett, A compendium of comparison function results, Math. Control Signals Syst., 26 (2014), 339-374.doi: 10.1007/s00498-014-0128-8.

    [132]

    C. Kellett, Classical converse theorems in Lyapunov's second method, Discrete Contin. Dyn. Syst. Ser. B, 8 (2015), 2333-2360.doi: 10.3934/dcdsb.2015.20.2333.

    [133]

    H. Khalil, Nonlinear Systems, Macmillan Publishing Company, New York, 1992.

    [134]

    V. Kharitonov, Time-delay Systems: Lyapunov Functionals and Matrices, Control Engineering. Birkhäuser/Springer, New York, 2013.doi: 10.1007/978-0-8176-8367-2.

    [135]

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

    [136]

    E. Kinnen and C. Chen, Liapunov functions derived from auxiliary exact differential equations, Automatica, 4 (1968), 195-204.doi: 10.1016/0005-1098(68)90014-9.

    [137]

    P. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Mathematical Society, 2011.doi: 10.1090/surv/176.

    [138]

    P. Koltai, A stochastic approach for computing the domain of attraction without trajectory simulation, in Dynamical Systems, Differential Equations and Applications, 8th AIMS Conference. Suppl., Vol. II, 2011, 854-863.

    [139]

    N. Krasovskiĭ, Problems of the Theory of Stability of Motion, English translation by Stanford University Press, 1963, Mir, Moskow, 1959.

    [140]

    A. Kravchuk, G. Leonov and D. Ponomarenko, A criterion for the strong orbital stability of the trajectories of dynamical systems I, Diff. Uravn., 28 (1992), 1507-1520.

    [141]

    V. Lakshmikantham, V. Matrosov and S. Sivasundaram, Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, Mathematics and its Applications, 63, Kluwer Academic Publishers Group, Dordrecht, 1991.doi: 10.1007/978-94-015-7939-1.

    [142]

    M. Lazar, On infinity norms as Lyapunov functions: Alternative necessary and sufficient conditions, in Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, USA, December 2010, 5936-5942.doi: 10.1109/CDC.2010.5717266.

    [143]

    M. Lazar and A. Doban, On infinity norms as Lyapunov functions for continuous-time dynamical systems, in Proceedings of the 50th IEEE Conference on Decision and Control, Orlando (Florida), USA, 2011, 7567-7572.doi: 10.1109/CDC.2011.6161163.

    [144]

    M. Lazar, A. Doban and N. Athanasopoulos, On stability analysis of discrete-time homogeneous dynamics, in Proceedings of the 17th International Conference on Systems Theory, Control and Computing, Sinaia, Romania, October 2013, 297-305.doi: 10.1109/ICSTCC.2013.6688976.

    [145]

    M. Lazar and A. Jokić, On infinity norms as Lyapunov functions for piecewise affine systems, in HSCC'10, Hybrid Systems: Computation and Control, ACM, New York, 2010, 131-139.doi: 10.1145/1755952.1755972.

    [146]

    G. Leonov, I. Burkin and A. Shepelyavyi, Frequency Methods in Oscillation Theory, Mathematics and its Applications, 357, Kluwer Academic Publishers Group, Dordrecht, 1996.doi: 10.1007/978-94-009-0193-3.

    [147]

    D. Lewis, Metric properties of differential equations, Amer. J. Math., 71 (1949), 294-312.doi: 10.2307/2372245.

    [148]

    H. Li, R. Baier, L. Grüne, S. Hafstein and F. Wirth, Computation of local ISS Lyapunov functions via linear programming, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS) (no. 0158), Groningen, The Netherlands, 2014, 1189-1195.

    [149]

    H. Li, R. Baier, L. Grüne, S. Hafstein and F. Wirth, Computation of local ISS Lyapunov functions with low gains via linear programming, Discrete Contin. Dyn. Syst. Ser. B, 8 (2015), 2477-2495.doi: 10.3934/dcdsb.2015.20.2477.

    [150]

    H. Li, S. Hafstein and C. Kellett, Computation of Lyapunov functions for discrete-time systems using the Yoshizawa construction, in Proceedings of the 53rd IEEE Conference on Decision and Control - CDC 2014, Los Angeles (CA), USA, 2014, 5512-5517.doi: 10.1109/CDC.2014.7040251.

    [151]

    D. Liberzon, Switching in Systems and Control, Systems & Control: Foundations & Applications, Birkhäuser, 2003.doi: 10.1007/978-1-4612-0017-8.

    [152]

    Y. Lin, E. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optimization, 34 (1996), 124-160.doi: 10.1137/S0363012993259981.

    [153]

    Z. Liu, The random case of Conley's theorem, Nonlinearity, 19 (2006), 277-291.doi: 10.1088/0951-7715/19/2/002.

    [154]

    Z. Liu, The random case of Conley's theorem. II. The complete Lyapunov function, Nonlinearity, 20 (2007), 1017-1030.doi: 10.1088/0951-7715/20/4/012.

    [155]

    Z. Liu, The random case of Conley's theorem. III. Random semiflow case and Morse decomposition, Nonlinearity, 20 (2007), 2773-2792.doi: 10.1088/0951-7715/20/12/003.

    [156]

    W. Lohmiller and J.-J. Slotine, On Contraction Analysis for Non-linear Systems, Automatica, 34 (1998), 683-696.doi: 10.1016/S0005-1098(98)00019-3.

    [157]

    K. Loparo and G. Blankenship, Estimating the domain of attraction of nonlinear feedback systems, IEEE Trans. Automat. Control, 23 (1978), 602-608.

    [158]

    A. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790; Translated by A. T. Fuller from Édouard Davaux's French translation (1907) of the 1892 Russian original, With an editorial (historical introduction) by Fuller, a biography of Lyapunov by V. I. Smirnov, and the bibliography of Lyapunov's works collected by J. F. Barrett, Lyapunov centenary issue.doi: 10.1080/00207179208934253.

    [159]

    M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Communications and Control Engineering, Springer, 2009.doi: 10.1007/978-1-84882-535-2.

    [160]

    I. Manchester and J.-J. Slotine, Transverse contraction criteria for existence, stability, and robustness of a limit cycle, Systems Control Lett., 63 (2014), 32-38.doi: 10.1016/j.sysconle.2013.10.005.

    [161]

    X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Woodhead Publishing, 2008.doi: 10.1533/9780857099402.

    [162]

    S. Margolis and W. Vogt, Control engineering applications of V. I. Zubov's construction procedure for Lyapunov functions, IEEE Trans. Automat. Control, 8 (1963), 104-113.doi: 10.1109/TAC.1963.1105553.

    [163]

    S. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming, Dynamical Systems: An International Journal, 17 (2002), 137-150.doi: 10.1080/0268111011011847.

    [164]

    S. Marinósson, Stability Analysis of Nonlinear Systems with Linear Programming: A Lyapunov Functions Based Approach, PhD thesis, Gerhard-Mercator-University Duisburg, Duisburg, Germany, 2002.

    [165]

    A. Martynyuk, Analysis of stability problems via Matrix Lyapunov Functions, J. Appl. Math. Stochastic Anal., 3 (1990), 209-226.doi: 10.1155/S104895339000020X.

    [166]

    J. Massera, On Liapounoff's conditions of stability, Ann. of Math., 50 (1949), 705-721.doi: 10.2307/1969558.

    [167]

    J. Massera, Contributions to stability theory, Ann. of Math., 64 (1956), 182-206; Erratum. Ann. of Math., 68 (1958), 202.doi: 10.2307/1969955.

    [168]

    V. Matrosov, On the stability of motion, J. Appl. Math. Mech., 26 (1963), 1337-1353.

    [169]

    P. Menck, J. Heitzig, N. Marwan and K. Kurths, How basin stability complements the linear-stability paradigm, Nature Physics, 9 (2013), 89-92.doi: 10.1038/nphys2516.

    [170]

    S. Meyn and R. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009.doi: 10.1017/CBO9780511626630.

    [171]

    A. Michel, L. Hou and D. Liu, Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems, Systems & Control: Foundations & Applications, Birkhäuser, 2008.

    [172]

    A. Michel, R. Miller and B. Nam, Stability analysis of interconnected systems using computer generated Lyapunov functions, IEEE Trans. Circuits and Systems, 29 (1982), 431-440.doi: 10.1109/TCS.1982.1085181.

    [173]

    A. Michel, B. Nam and V. Vittal, Computer generated Lyapunov functions for interconnected systems: Improved results with applications to power system, IEEE Trans. Circuits and Systems, 31 (1984), 189-198.doi: 10.1109/TCS.1984.1085483.

    [174]

    A. Michel, N. Sarabudla and R. Miller, Stability analysis of complex dynamical systems, Circuits Systems Signal Process, 1 (1982), 171-202.doi: 10.1007/BF01600051.

    [175]

    C. Mikkelsen, Numerical Methods For Large Lyapunov Equations, PhD thesis, Purdue University, West Lafayette (IN), USA, 2009.

    [176]

    N. Mohammed and P. Giesl, Grid refinement in the construction of Lyapunov functions using radial basis functions, Discrete Contin. Dyn. Syst. Ser. B, 8 (2015), 2453-2476.doi: 10.3934/dcdsb.2015.20.2453.

    [177]

    A. Molchanov and E. Pyatnitskiiĭ, Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Systems Control Lett., 13 (1989), 59-64.doi: 10.1016/0167-6911(89)90021-2.

    [178]

    A. Molchanov and E. Pyatnitskiĭ, Lyapunov functions that specify necessary and sufficient conditions of absolute stability of nonlinear nonstationary control systems I, II, Automat. Remote Control, 47 (1986), 344-354, 443-451.

    [179]

    N. Noroozi, P. Karimaghaee, F. Safaei and H. Javadi, Generation of Lyapunov functions by neural networks, in Proceedings of the World Congress on Engineering 2008, 2008.

    [180]

    D. Norton, The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolinae, 36 (1995), 585-597.

    [181]

    Y. Ohta, On the construction of piecewise linear Lyapunov functions, in Proceedings of the 40th IEEE Conference on Decision and Control, 3 (2001), 2173-2178.doi: 10.1109/CDC.2001.980577.

    [182]

    Y. Ohta and M. Tsuji, A generalization of piecewise linear Lyapunov functions, in Proceedings of the 42nd IEEE Conference on Decision and Control, 5 (2003), 5091-5096.doi: 10.1109/CDC.2003.1272443.

    [183]

    R. O'Shea, The extension of Zubov's method to sampled data control systems described by nonlinear autonomous difference equations, IEEE Trans. Automat. Control, 9 (1964), 62-70.

    [184]

    S. Panikhom and S. Sujitjorn, Numerical approach to construction of Lyapunov function for nonlinear stability analysis, Research Journal of Applied Sciences, Engineering and Technology, 4 (2012), 2915-2919.

    [185]

    A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Pranja, P. Seiler and P. Parrilo, SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB, User's Guide, Version 3.00 edition, 2013.

    [186]

    P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiziation, PhD thesis, California Institute of Technology Pasadena, California, 2000.

    [187]

    M. Patrão, Existence of complete Lyapunov functions for semiflows on separable metric spaces, Far East J. Dyn. Syst., 17 (2011), 49-54.

    [188]

    M. Peet, Exponentially stable nonlinear systems have polynomial Lyapunov functions on bounded regions, IEEE Trans. Automat. Control, 54 (2009), 979-987.doi: 10.1109/TAC.2009.2017116.

    [189]

    M. Peet and A. Papachristodoulou, A converse sum of squares Lyapunov result with a degree bound, IEEE Trans. Automat. Control, 57 (2012), 2281-2293.doi: 10.1109/TAC.2012.2190163.

    [190]

    S. Pettersson and B. Lennartson, Stability and robustness for hybrid systems, in Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, 1996, 1202-1207.doi: 10.1109/CDC.1996.572653.

    [191]

    A. Polanski, Lyapunov functions construction by linear programming, IEEE Trans. Automat. Control, 42 (1997), 1113-1116.doi: 10.1109/9.599986.

    [192]

    A. Polanski, On absolute stability analysis by polyhedral Lyapunov functions, Automatica, 36 (2000), 573-578.doi: 10.1016/S0005-1098(99)00180-6.

    [193]

    I. Pólik and T. Terlaky, A survey of the S-lemma, SIAM Review, 49 (2007), 371-418.doi: 10.1137/S003614450444614X.

    [194]

    C. Prieur and F. Mazenc, ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws, Math. Control Signals Syst., 24 (2012), 111-134.doi: 10.1007/s00498-012-0074-2.

    [195]

    D. Prokhorov, A Lyapunov machine for stability analysis of nonlinear systems, in Proceedings of the IEEE International Conference on Neural Networks, Vol. 2, Orlando (FL), USA, 1994, 1028-1031.doi: 10.1109/ICNN.1994.374324.

    [196]

    S. Raković and M. Lazar, The Minkowski-Lyapunov equation for linear dynamics: Theoretical foundations, Automatica, 50 (2014), 2015-2024.doi: 10.1016/j.automatica.2014.05.023.

    [197]

    M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, 1907, Springer, Berlin, 2007.

    [198]

    M. Rasmussen, Morse decompositions of nonautonomous dynamical systems, Trans. Amer. Math. Soc., 359 (2007), 5091-5115.doi: 10.1090/S0002-9947-07-04318-8.

    [199]

    S. Ratschan and Z. She, Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions, SIAM J. Control Optim., 48 (2010), 4377-4394.doi: 10.1137/090749955.

    [200]

    M. Rezaiee-Pajand and B. Moghaddasie, Estimating the region of attraction via collocation for autonomous nonlinear systems, Structural Engineering and Mechanics, 41 (2012), 263-284.

    [201]

    M. Roozbehani, S. Megretski and E. Feron, Optimization of Lyapunov invariants in verification of software systems, IEEE Trans. Automat. Control, 58 (2013), 696-711.doi: 10.1109/TAC.2013.2241472.

    [202]

    B. Rüffer, N. van de Wouw and M. Mueller, Convergent systems vs. incremental stability, Systems Control Lett., 62 (2013), 277-285.doi: 10.1016/j.sysconle.2012.11.015.

    [203]

    G. Serpen, Empirical approximation for Lyapunov functions with artificial neural nets, in Proceedings of the 2005 IEEE International Joint Conference on Neural Networks, Vol. 2, Montreal (QC), Canada, 2005, 735-740.doi: 10.1109/IJCNN.2005.1555943.

    [204]

    Z. She, H. Li, B. Xue, Z. Zheng and B. Xia, Discovering polynomial Lyapunov functions for continuous dynamical systems, J. Symbolic Comput., 58 (2013), 41-63.doi: 10.1016/j.jsc.2013.06.003.

    [205]

    Z. She and B. Xue, Computing an invariance kernel with target by computing Lyapunov-like functions, IET Control Theory Appl., 7 (2013), 1932-1940.doi: 10.1049/iet-cta.2013.0275.

    [206]

    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.

    [207]

    D. Šiljak, Large-scale Dynamic Systems. Stability and Structure, North-Holland Series in System Science and Engineering, 3, North-Holland Publishing Co., New York-Amsterdam, 1979.

    [208]

    E. Sontag, A Lyapunov-like characterization of asymptotic controllability, SIAM J. Control Optimization, 21 (1983), 462-471.doi: 10.1137/0321028.

    [209]

    E. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.doi: 10.1109/9.28018.

    [210]

    E. Sontag, New characterizations of input-to-state stability, IEEE Trans. Automat. Control, 41 (1996), 1283-1294.doi: 10.1109/9.536498.

    [211]

    E. Sontag, Mathematical Control Theory, 2nd edition, Springer, 1998.doi: 10.1007/978-1-4612-0577-7.

    [212]

    E. Sontag and H. Sussman, Nonsmooth control-Lyapunov functions, in Proceedings of the 34th IEEE Conference on Decision and Control, Vol. 3, New Orleans (LA), USA, 1995, 2799-2805.doi: 10.1109/CDC.1995.478542.

    [213]

    E. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Systems Control Lett., 24 (1995), 351-359.doi: 10.1016/0167-6911(94)00050-6.

    [214]

    B. Stenström, Dynamical systems with a certain local contraction property, Math. Scand., 11 (1962), 151-155.

    [215]

    A. Subbaraman and A. Teel, A converse Lyapunov theorem for strong global recurrence, Automatica, 49 (2013), 2963-2974.doi: 10.1016/j.automatica.2013.07.001.

    [216]

    A. Subbaraman and A. Teel, A Matrosov theorem for strong global recurrence, Automatica, 49 (2013), 3390-3395.doi: 10.1016/j.automatica.2013.08.009.

    [217]

    Z. Sun, Stability of piecewise linear systems revisited, Annu. Rev. Control, 34 (2010), 221-231.doi: 10.1016/j.arcontrol.2010.08.003.

    [218]

    Z. Sun and S. Ge, Stability Theory of Switched Dynamical Systems, Communications and Control Engineering, Springer, 2011.doi: 10.1007/978-0-85729-256-8.

    [219]

    K. Tanaka, T. Hori and H. Wang, A multiple Lyapunov function approach to stabilization of fuzzy control systems, IEEE T. Fuzzy Syst., 11 (2003), 582-589.doi: 10.1109/TFUZZ.2003.814861.

    [220]

    A. R. Teel and L. Praly, A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions, ESAIM Control Optim. Calc. Var., 5 (2000), 313-367.doi: 10.1051/cocv:2000113.

    [221]

    R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Springer, 1997.doi: 10.1007/978-1-4612-0645-3.

    [222]

    A. Tesi, F. Villoresi and R. Genesio, On stability domain estimation via a quadratic Lyapunov function: Convexity and optimality properties for polynomial systems, in Proceedings of the 33rd Conference on Decision and Control, Lake Buena Vista (FL), USA, 1994, 1907-1912.doi: 10.1109/CDC.1994.411100.

    [223]

    A. Vannelli and M. Vidyasagar, Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems, Automatica, 21 (1985), 69-80.doi: 10.1016/0005-1098(85)90099-8.

    [224]

    K. Wang and A. Michel, On the stability of a family of nonlinear time-varying system, IEEE Trans. Circuits and Systems, 43 (1996), 517-531.doi: 10.1109/81.508171.

    [225]

    H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005.

    [226]

    C. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems, ESAIM Control Optim. Calc. Var., 7 (2002), 421-442.doi: 10.1051/cocv:2002062.

    [227]

    C. Yfoulis and R. Shorten, A numerical technique for the stability analysis of linear switched systems, Int. J. Control, 77 (2004), 1019-1039.doi: 10.1080/002071704200026963.

    [228]

    T. Yoshizawa, Stability Theory by Liapunov's Second Method, Publications of the Mathematical Society of Japan, No. 9. The Mathematical Society of Japan, Tokyo, 1966.

    [229]

    V. Zubov, Methods of A. M. Lyapunov and Their Application, Translation prepared under the auspices of the United States Atomic Energy Commission; edited by Leo F. Boron, P. Noordhoff Ltd, Groningen, 1964.

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