Review on computational methods for Lyapunov functions. (English) Zbl 1337.37001
In 1892 Lyapunov published his doctoral dissertation in which he introduced a sufficient condition for the stability of a nonlinear system, namely, the existence of a positive definite function decreasing along the solution trajectories. These functions are now known as Lyapunov functions. This review paper is divided in two parts, the first discussing different Lyapunov and Lyapunov-type functions for various types of systems, and the second discussing numerical methods for computing Lyapunov functions. Besides finite-dimensional systems of ordinary differential equations, Lyapunov functions have been developed for discrete-time systems (maps), PDEs, non-autonomous systems of ODEs, switched systems, delay differential equations, control systems and random dynamical systems, among others. Generalisations of Lyapunov functions include non-smooth Lyapunov functions, and vector, matrix and multiple Lyapunov functions.
Numerical methods used to compute Lyapunov functions include collocation, linear programming, linear matrix inequalities, algebraic methods and graph theoretic methods. In this review the authors bring together these different methods and describe the state of the art for the vast variety of methods to compute Lyapunov functions for various kinds of systems.
Numerical methods used to compute Lyapunov functions include collocation, linear programming, linear matrix inequalities, algebraic methods and graph theoretic methods. In this review the authors bring together these different methods and describe the state of the art for the vast variety of methods to compute Lyapunov functions for various kinds of systems.
Reviewer: Carlo Laing (Auckland)
MSC:
| 37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |
| 37M99 | Approximation methods and numerical treatment of dynamical systems |
| 34D20 | Stability of solutions to ordinary differential equations |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 37C75 | Stability theory for smooth dynamical systems |
| 34D45 | Attractors of solutions to ordinary differential equations |
| 37B25 | Stability of topological dynamical systems |
Keywords:
Lyapunov function; stability; basin of attraction; dynamical system; contraction metric; converse theorem; numerical methodReferences:
| [1] | N. Aghannan, An intrinsic observer for a class of Lagrangian systems,, IEEE Trans. Automat. Control, 48, 936 (2003) · Zbl 1364.93078 · doi:10.1109/TAC.2003.812778 |
| [2] | A. Agrachev, Lie-algebraic stability criteria for switched systems,, SIAM J. Control Optim., 40, 253 (2001) · Zbl 0995.93064 · doi:10.1137/S0363012999365704 |
| [3] | A. Ahmadi, On complexity of Lyapunov functions for switched linear systems,, in Proceedings of the 19th World Congress of the International Federation of Automatic Control (2014) |
| [4] | A. Ahmadi, A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function,, in Proceedings of the 50th IEEE Conference on Decision and Control (CDC), 7579 (2011) · doi:10.1109/CDC.2011.6161499 |
| [5] | A. Ahmadi, Complexity of ten decision problems in continuous time dynamical systems,, in Proceedings of the American Control Conference, 6376 (2013) · doi:10.1109/ACC.2013.6580838 |
| [6] | E. Akin, <em>The General Topology of Dynamical Systems</em>,, American Mathematical Society (1993) · Zbl 0781.54025 |
| [7] | A. Aleksandrov, Professor V. I. Zubov to the 80th birthday anniversary,, Nonlinear Dyn. Syst. Theory, 10, 1 (2010) |
| [8] | R. Ambrosino, 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, 2872 (2012) · doi:10.1109/CDC.2012.6427016 |
| [9] | J. Anderson, Advances in computational Lyapunov analysis using sum-of-squares programming,, Discrete Contin. Dyn. Syst. Ser. B, 8, 2361 (2015) · Zbl 1334.37105 · doi:10.3934/dcdsb.2015.20.2361 |
| [10] | D. Angeli, A Lyapunov approach to incremental stability properties,, IEEE Trans. Automat. Contr., 47, 410 (2002) · Zbl 1364.93552 · doi:10.1109/9.989067 |
| [11] | E. Aragão-Costa, Stability of gradient semigroups under perturbations,, Nonlinearity, 24, 2099 (2011) · Zbl 1225.37018 · doi:10.1088/0951-7715/24/7/010 |
| [12] | E. Aragão-Costa, Non-autonomous Morse-decomposition and Lyapunov functions for gradient-like processes,, Trans. Amer. Math. Soc., 365, 5277 (2013) · Zbl 1291.37002 · doi:10.1090/S0002-9947-2013-05810-2 |
| [13] | L. Arnold, <em>Stochastic Differential Equations: Theory and Applications</em>,, Wiley (1974) · Zbl 0278.60039 |
| [14] | L. Arnold, Random dynamical systems,, in Dynamical Systems (Montecatini Terme, 1 (1994) · Zbl 0805.60048 · doi:10.1007/BFb0095238 |
| [15] | L. Arnold, Lyapunov’s second method for random dynamical systems,, J. Differential Equations, 177, 235 (2001) · Zbl 1040.37035 · doi:10.1006/jdeq.2000.3991 |
| [16] | J.-P. Aubin, <em>Differential Inclusions</em>,, 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, 1431 (1983) · Zbl 0532.34035 · doi:10.1016/0362-546X(83)90010-X |
| [18] | E. Aylward, Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming,, Automatica, 44, 2163 (2008) · Zbl 1283.93217 · doi:10.1016/j.automatica.2007.12.012 |
| [19] | R. Baier, Linear programming based Lyapunov function computation for differential inclusions,, Discrete Contin. Dyn. Syst. Ser. B, 17, 33 (2012) · Zbl 1235.93222 · doi:10.3934/dcdsb.2012.17.33 |
| [20] | R. Baier, 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), 1173 (0232) |
| [21] | H. Ban, A computational approach to Conley’s decomposition theorem,, J. Comput. Nonlinear Dynam., 1, 312 (2006) · doi:10.1115/1.2338651 |
| [22] | E. Barbašin, On the existence of Lyapunov functions in the case of asymptotic stability in the large,, Prikl. Mat. Meh., 18, 345 (1954) |
| [23] | R. Bartels, Solution of the matrix equation AX+XB=C,, Communications of the ACM, 15, 820 (1972) · Zbl 1372.65121 · doi:10.1145/361573.361582 |
| [24] | R. Bellman, Vector Lyapunov functions,, J. SIAM Control Ser. A, 1, 32 (1962) · Zbl 0144.10901 |
| [25] | R. Bellman, <em>Introduction to Matrix Analysis</em>,, Classics in Applied Mathematics (1995) · Zbl 0824.15002 |
| [26] | A. Berger, On finite-time hyperbolicity,, Commun. Pure Appl. Anal., 10, 963 (2011) · Zbl 1239.34048 · doi:10.3934/cpaa.2011.10.963 |
| [27] | A. Berger, A definition of spectrum for differential equations on finite time,, J. Differential Equations, 246, 1098 (2009) · Zbl 1169.34040 · doi:10.1016/j.jde.2008.06.036 |
| [28] | J. Bernussou, A linear programming oriented procedure for quadratic stabilization of uncertain systems,, Systems Control Lett., 13, 65 (1989) · Zbl 0678.93042 · doi:10.1016/0167-6911(89)90022-4 |
| [29] | N. Bhatia, <em>Dynamical Systems: Stability Theory and Applications</em>,, Lecture Notes in Mathematics (1967) · Zbl 0155.42201 |
| [30] | G. Birkhoff, <em>Dynamical Systems</em>,, American Mathematical Society Colloquium Publications (1966) · Zbl 0171.05402 |
| [31] | J. Björnsson, Algorithmic verification of approximations to complete Lyapunov functions,, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (no. 0180), 1181 (0180) |
| [32] | J. Björnsson, Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction,, in Proceedings of the CDC, 5506 (2014) |
| [33] | J. Björnsson, Computation of Lyapunov functions for systems with multiple attractors,, Discrete Contin. Dyn. Syst. Ser. A, 35, 4019 (2015) · Zbl 1366.37028 · 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, 1755 (1991) · Zbl 0800.93754 · 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, 428 (1994) · Zbl 0800.93754 · doi:10.1109/9.272351 |
| [36] | F. Blanchini, Nonquadratic Lyapunov functions for robust control,, Automatica, 31, 451 (1995) · Zbl 0825.93653 · doi:10.1016/0005-1098(94)00133-4 |
| [37] | F. Blanchini, 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, 1105 (1994) · doi:10.1109/CDC.1994.411291 |
| [38] | F. Blanchini, <em>Set-theoretic Methods in Control</em>,, Systems & Control: Foundations & Applications (2008) · Zbl 1140.93001 |
| [39] | V. Boichenko, <em>Dimension Theory for Ordinary Differential Equations</em>,, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics] (2005) · Zbl 1094.34002 · doi:10.1007/978-3-322-80055-8 |
| [40] | G. Borg, <em>A Condition for the Existence Of Orbitally Stable Solutions of Dynamical Systems</em>,, Kungliga Tekniska Högskolan Handlingar Stockholm (1960) · Zbl 0091.08402 |
| [41] | J. Bouvrie, Model reduction for nonlinear control systems using kernel subspace methods,, <a href= (2011) |
| [42] | S. Boyd, <em>Linear Matrix Inequalities in System and Control Theory</em>,, SIAM Studies in Applied Mathematics (1994) · Zbl 0816.93004 · doi:10.1137/1.9781611970777 |
| [43] | S. Boyd, <em>Convex Optimization</em>,, Cambridge University Press (2004) · Zbl 1058.90049 · doi:10.1017/CBO9780511804441 |
| [44] | M. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,, IEEE Trans. Automat. Control, 43, 475 (1998) · Zbl 0904.93036 · doi:10.1109/9.664150 |
| [45] | R. Brayton, Stability of dynamical systems: A constructive approach,, IEEE Trans. Circuits and Systems, 26, 224 (1979) · Zbl 0413.93048 · doi:10.1109/TCS.1979.1084637 |
| [46] | R. Brayton, Constructive stability and asymptotic stability of dynamical systems,, IEEE Trans. Circuits and Systems, 27, 1121 (1980) · Zbl 0458.93047 · doi:10.1109/TCS.1980.1084749 |
| [47] | M. Buhmann, <em>Radial Basis Functions: Theory and Implementations</em>,, Cambridge Monographs on Applied and Computational Mathematics (2003) · Zbl 1038.41001 · doi:10.1017/CBO9780511543241 |
| [48] | H. Burchardt, 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, 241 (2007) |
| [49] | C. Byrnes, Topological methods for nonlinear oscillations,, Notices Amer. Math. Soc., 57, 1080 (2010) · Zbl 1203.37033 |
| [50] | F. Camilli, A generalization of Zubov’s method to perturbed systems,, SIAM J. Control Optim., 40, 496 (2001) · Zbl 1006.93061 · doi:10.1137/S036301299936316X |
| [51] | F. Camilli, A regularization of Zubov’s equation for robust domains of attraction,, in Nonlinear Control in the Year 2000, 277 (2000) · Zbl 1239.93048 · doi:10.1007/BFb0110220 |
| [52] | F. Camilli, Control Lyapunov functions and Zubov’s method,, SIAM J. Control Optim., 47, 301 (2008) · Zbl 1157.93028 · doi:10.1137/06065129X |
| [53] | A. Carvalho, <em>Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems</em>,, Applied Mathematical Sciences (2013) · Zbl 1263.37002 · doi:10.1007/978-1-4614-4581-4 |
| [54] | C. Chen, Construction of Liapunov functions,, J. Franklin Inst., 289, 133 (1970) · Zbl 0304.34049 · 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, 2500 (2010) · Zbl 1368.93496 · doi:10.1109/TAC.2010.2046926 |
| [56] | C. Chicone, <em>Ordinary Differential Equations with Applications</em>,, Texts in Applied Mathematics (1999) · Zbl 0937.34001 |
| [57] | I. Chueshov, <em>Introduction to the Theory of Infinite-Dimensional Dissipative Systems</em>,, ACTA Scientific Publishing House (2002) · Zbl 1100.37047 |
| [58] | F. Clarke, Lyapunov functions and discontinuous stabilizing feedback,, Annu. Rev. Control, 35, 13 (2011) · doi:10.1016/j.arcontrol.2011.03.001 |
| [59] | F. Clarke, Asymptotic stability and smooth Lyapunov functions,, J. Differential Equations, 149, 69 (1998) · Zbl 0907.34013 · doi:10.1006/jdeq.1998.3476 |
| [60] | C. Conley, <em>Isolated Invariant Sets and the Morse Index</em>,, CBMS Regional Conference Series (1978) · Zbl 0397.34056 |
| [61] | J.-M. Coron, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, IEEE Trans. Automat. Control, 52, 2 (2007) · Zbl 1366.93481 · doi:10.1109/TAC.2006.887903 |
| [62] | E. Davison, A computational method for determining quadratic Lyapunov functions for non-linear systems,, Automatica, 7, 627 (1971) · Zbl 0225.34027 · doi:10.1016/0005-1098(71)90027-6 |
| [63] | G. Davrazos, A review of stability results for switched and hybrid systems,, in Proceedings of 9th Mediterranean Conference on Control and Automation (2001) |
| [64] | W. Dayawansa, A converse Lyapunov theorem for a class of dynamical systems which undergo switching,, IEEE Tra, 44, 751 (1999) · Zbl 0960.93046 · doi:10.1109/9.754812 |
| [65] | M. Dellnitz, The algorithms behind {GAIO} - set oriented numerical methods for dynamical systems,, in Ergodic Theory, 145 (2001) · Zbl 0998.65126 |
| [66] | M. Dellnitz, Set oriented numerical methods for dynamical systems,, in Handbook of Dynamical Systems, 221 (2002) · Zbl 1036.37030 · doi:10.1016/S1874-575X(02)80026-1 |
| [67] | U. Dini, <em>Fondamenti per la Teoria Delle Funzioni di Variabili Reali</em>,, (in Italian) Pisa (1878) · JFM 10.0274.01 |
| [68] | S. Dubljević, A new Lyapunov design approach for nonlinear systems based on Zubov’s method,, Automatica, 38, 1999 (2002) · Zbl 1011.93089 · doi:10.1016/S0005-1098(02)00110-3 |
| [69] | N. Eghbal, Discontinuous piecewise quadratic Lyapunov functions for planar piecewise affine systems,, J. Math. Anal. Appl., 399, 586 (2013) · Zbl 1254.93126 · doi:10.1016/j.jmaa.2012.09.054 |
| [70] | K. Erickson, 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, 113 (1985) · Zbl 0557.93056 · doi:10.1109/TCS.1985.1085676 |
| [71] | K. Erickson, 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, 132 (1985) · Zbl 0557.93057 · 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 (1997) |
| [73] | F. Fallside, Control engineering applications of V. I. Zubov’s construction procedure for Lyapunov functions,, IEEE Trans. Automat. Control, 10, 220 (1965) · doi:10.1109/TAC.1965.1098103 |
| [74] | F. Faria, Reducing the conservatism of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy Lyapunov functions,, International Journal of Systems Science, 44, 1956 (2013) · Zbl 1307.93218 · doi:10.1080/00207721.2012.670307 |
| [75] | D. R. Ferguson, Generalisation of Zubov’s construction procedure for Lyapunov functions,, Electron. Lett., 6, 73 (1970) · doi:10.1049/el:19700046 |
| [76] | A. Filippov, <em>Differential Equations with Discontinuous Right-hand Side</em>,, Translated from Russian (1985) · Zbl 0571.34001 |
| [77] | H. Flashner, A computational approach for studying domains of attraction for nonlinear systems,, Internat. J. Non-Linear Mech., 23, 279 (1988) · Zbl 0655.70019 · doi:10.1016/0020-7462(88)90026-1 |
| [78] | F. Forni, A differential Lyapunov framework for Contraction Analysis,, IEEE Trans. Automat. Control, 59, 614 (2014) · Zbl 1360.93597 · 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, 1, 798 (1991) · doi:10.1109/CDC.1991.261424 |
| [80] | R. Geiselhart, An alternative converse Lyapunov theorem for discrete-time systems,, Systems Control Lett., 70, 49 (2014) · Zbl 1290.93149 · doi:10.1016/j.sysconle.2014.05.007 |
| [81] | R. Genesio, On the estimation of asymptotic stability regions: State of the art and new proposals,, IEEE Trans. Automat. Control, 30, 747 (1985) · Zbl 0568.93054 · doi:10.1109/TAC.1985.1104057 |
| [82] | P. Giesl, Necessary conditions for a limit cycle and its basin of attraction,, Nonlinear Anal., 56, 643 (2004) · Zbl 1048.34064 · 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, 89 (2005) · Zbl 1084.34005 · doi:10.1002/zamm.200310164 |
| [84] | P. Giesl, <em>Construction of Global Lyapunov Functions Using Radial Basis Functions</em>,, Lecture Notes in Math. (1904) |
| [85] | P. Giesl, On the determination of the basin of attraction of discrete dynamical systems,, J. Difference Equ. Appl., 13, 523 (2007) · Zbl 1120.39018 · doi:10.1080/10236190601135209 |
| [86] | P. Giesl, Construction of a local and global Lyapunov function using radial basis functions,, IMA J. Appl. Math., 73, 782 (2008) · Zbl 1155.34030 · 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, 606 (2009) · Zbl 1173.34034 · 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, 2387 (2012) · Zbl 1254.34072 · doi:10.3934/dcdsb.2012.17.2387 |
| [89] | P. Giesl, Converse theorems on contraction metrics for an equilibrium,, J. Math. Anal. Appl., 424, 1380 (2015) · Zbl 1331.34120 · doi:10.1016/j.jmaa.2014.12.010 |
| [90] | P. Giesl, Existence of piecewise affine Lyapunov functions in two dimensions,, J. Math. Anal. Appl., 371, 233 (2010) · Zbl 1205.34064 · doi:10.1016/j.jmaa.2010.05.009 |
| [91] | P. Giesl, Existence of piecewise linear Lyapunov functions in arbitrary dimensions,, Discrete Contin. Dyn. Syst., 32, 3539 (2012) · Zbl 1266.37010 · doi:10.3934/dcds.2012.32.3539 |
| [92] | P. Giesl, Construction of a CPA contraction metric for periodic orbits using semidefinite optimization,, Nonlinear Anal., 86, 114 (2013) · Zbl 1294.34046 · doi:10.1016/j.na.2013.03.012 |
| [93] | P. Giesl, Computation of Lyapunov functions for nonlinear discrete time systems by linear programming,, J. Difference Equ. Appl., 20, 610 (2014) · Zbl 1319.39010 · doi:10.1080/10236198.2013.867341 |
| [94] | P. Giesl, Revised CPA method to compute Lyapunov functions for nonlinear systems,, J. Math. Anal. Appl., 410, 292 (2014) · Zbl 1317.34134 · doi:10.1016/j.jmaa.2013.08.014 |
| [95] | P. Giesl, Areas of attraction for nonautonomous differential equations on finite time intervals,, J. Math. Anal. Appl., 390, 27 (2012) · Zbl 1239.34058 · doi:10.1016/j.jmaa.2011.12.051 |
| [96] | P. Giesl, Meshless collocation: Error estimates with application to dynamical systems,, SIAM J. Numer. Anal., 45, 1723 (2007) · Zbl 1148.65090 · doi:10.1137/060658813 |
| [97] | P. Giesl, Approximating the basin of attraction of time-periodic {ODE}s by meshless collocation,, Discrete Contin. Dyn. Syst., 25, 1249 (2009) · Zbl 1195.34076 · doi:10.3934/dcds.2009.25.1249 |
| [98] | P. Giesl, Numerical determination of the basin of attraction for asymptotically autonomous dynamical systems,, Nonlinear Anal., 75, 2823 (2012) · Zbl 1246.37045 · doi:10.1016/j.na.2011.11.027 |
| [99] | R. Goebel, <em>Hybrid Dynamical Systems</em>,, Modeling (2012) · Zbl 1241.93002 |
| [100] | R. Goebel, Conjugate convex Lyapunov functions for dual linear differential inclusions,, IEEE Trans. Automat. Control, 51, 661 (2006) · Zbl 1366.34030 · doi:10.1109/TAC.2006.872764 |
| [101] | A. Goullet, Efficient computation of Lyapunov functions for Morse decompositions,, Discrete Contin. Dyn. Syst. Ser. B, 8, 2419 (2015) · Zbl 1366.37031 · doi:10.3934/dcdsb.2015.20.2419 |
| [102] | B. Grosman, Lyapunov-based stability analysis automated by genetic programming,, Automatica, 45, 252 (2009) · Zbl 1154.93405 · 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, 1871 (1992) · Zbl 0771.93075 · 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, 971 (1994) · Zbl 0801.34052 · doi:10.1016/0362-546X(94)90060-4 |
| [105] | L. Grüne, <em>Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization</em>,, Lecture Notes in Mathematics (1783) · Zbl 0991.37001 · doi:10.1007/b83677 |
| [106] | L. Grüne, Lyapunov’s second method for nonautonomous differential equations,, Discrete Contin. Dyn. Syst., 18, 375 (2007) · Zbl 1128.37010 · doi:10.3934/dcds.2007.18.375 |
| [107] | \(O. \ddot G\) urel, A guide to the generation of Liapunov functions,, Indust. Engrg. Chem., 61, 30 (1969) · doi:10.1021/ie50711a006 |
| [108] | S. Hafstein, A constructive converse Lyapunov theorem on exponential stability,, Discrete Contin. Dyn. Syst., 10, 657 (2004) · Zbl 1070.93044 · 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, 281 (2005) · Zbl 1102.34038 · doi:10.1080/14689360500164873 |
| [110] | S. Hafstein, <em>An Algorithm for Constructing Lyapunov Functions</em>,, Electronic Journal of Differential Equations. Monograph (2007) · Zbl 1131.37021 |
| [111] | S. Hafstein, Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction,, in Proceedings of the 2014 American Control Conference (no. 0170), 548 (2014) · doi:10.1109/ACC.2014.6858660 |
| [112] | W. Hahn, <em>Stability of Motion</em>,, Springer (1967) · Zbl 0189.38503 |
| [113] | G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields,, Chaos, 10, 99 (2000) · Zbl 0979.37012 · doi:10.1063/1.166479 |
| [114] | B. Hargrave, <em>Using Genetic Algorithms to Optimize Control Lyapunov Functions</em>,, PhD thesis (2008) |
| [115] | P. Hartman, <em>Ordinary Differential Equations</em>,, Wiley (1964) · Zbl 0125.32102 |
| [116] | P. Hartman, On global asymptotic stability of solutions of differential equations,, Trans. Amer. Math. Soc., 104, 154 (1962) · Zbl 0109.06003 |
| [117] | M. Abu Hassan, Numerical determination of domains of attraction for electrical power systems using the method of Zubov,, Int. J. Control, 34, 371 (1981) · Zbl 0466.93021 · doi:10.1080/00207178108922536 |
| [118] | C. Hsu, <em>Cell-to-cell Mapping</em>,, Applied Mathematical Sciences (1987) · Zbl 0632.58002 · doi:10.1007/978-1-4757-3892-6 |
| [119] | T. Hu, Composite quadratic Lyapunov functions for constrained control systems,, IEEE Trans. Automat. Control, 48, 440 (2003) · Zbl 1364.93108 · doi:10.1109/TAC.2003.809149 |
| [120] | M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces,, Proc. Amer. Math. Soc., 126, 245 (1998) · Zbl 0891.58025 · doi:10.1090/S0002-9939-98-04500-6 |
| [121] | B. Ingalls, An infinite-time relaxation theorem for differential inclusions,, Proc. Amer. Math. Soc., 131, 487 (2003) · Zbl 1020.34016 · doi:10.1090/S0002-9939-02-06539-5 |
| [122] | T. Johansen, Computation of Lyapunov functions for smooth, nonlinear systems using convex optimization,, Automatica, 36, 1617 (2000) · Zbl 0971.93069 · doi:10.1016/S0005-1098(00)00088-1 |
| [123] | M. Johansson, <em>Piecewise Linear Control Systems</em>,, Lecture Notes in Control and Information Sciences (2003) · Zbl 1008.93002 · doi:10.1007/3-540-36801-9 |
| [124] | M. Johansson, Computation of piecewise quadratic Lyapunov functions for hybrid systems,, IEEE Trans. Automat. Control, 43, 555 (1998) · Zbl 0905.93039 · doi:10.1109/9.664157 |
| [125] | P. Julian, <em>A High Level Canonical Piecewise Linear Representation: Theory and Applications</em>,, PhD thesis (1999) |
| [126] | P. Julian, A parametrization of piecewise linear Lyapunov functions via linear programming,, Int. J. Control, 72, 702 (1999) · Zbl 0963.93036 · doi:10.1080/002071799220876 |
| [127] | O. Junge, Rigorous discretization of subdivision techniques,, in International Conference on Differential Equations, 916 (1999) · Zbl 0963.65536 |
| [128] | W. Kalies, An algorithmic approach to chain recurrence,, Found. Comput. Math, 5, 409 (2005) · Zbl 1099.37010 · doi:10.1007/s10208-004-0163-9 |
| [129] | R. Kamyar, Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares,, Discrete Contin. Dyn. Syst. Ser. B, 2383 · Zbl 1332.93291 · doi:10.3934/dcdsb.2015.20.2383 |
| [130] | J. Kapinski, Simulation-guided Lyapunov analysis for hybrid dynamical systems,, in Proceedings of the 17th International Conference on Hybrid Systems: Computation and Control (HSCC 2014), 133 (2014) · Zbl 1362.93108 · doi:10.1145/2562059.2562139 |
| [131] | C. Kellett, A compendium of comparison function results,, Math. Control Signals Syst., 26, 339 (2014) · Zbl 1294.93071 · 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, 2333 (2015) · Zbl 1332.93292 · doi:10.3934/dcdsb.2015.20.2333 |
| [133] | H. Khalil, <em>Nonlinear Systems</em>,, Macmillan Publishing Company (1992) · Zbl 0969.34001 |
| [134] | V. Kharitonov, <em>Time-delay Systems: Lyapunov Functionals and Matrices</em>,, Control Engineering. Birkhäuser/Springer (2013) · Zbl 1285.93071 · doi:10.1007/978-0-8176-8367-2 |
| [135] | R. Khasminskii, <em>Stochastic Stability of Differential Equations</em>,, Springer (2012) · Zbl 1241.60002 · doi:10.1007/978-3-642-23280-0 |
| [136] | E. Kinnen, Liapunov functions derived from auxiliary exact differential equations,, Automatica, 4, 195 (1968) · Zbl 0155.13701 · doi:10.1016/0005-1098(68)90014-9 |
| [137] | P. Kloeden, <em>Nonautonomous Dynamical Systems</em>,, Amer. Mathematical Society (2011) · Zbl 1244.37001 · doi:10.1090/surv/176 |
| [138] | P. Koltai, A stochastic approach for computing the domain of attraction without trajectory simulation,, in Dynamical Systems, 854 (2011) · Zbl 1306.37096 |
| [139] | N. Krasovskiĭ, <em>Problems of the Theory of Stability of Motion</em>,, English translation by Stanford University Press (1963) · Zbl 0109.06001 |
| [140] | A. Kravchuk, A criterion for the strong orbital stability of the trajectories of dynamical systems I,, Diff. Uravn., 28, 1507 (1992) · Zbl 0810.34043 |
| [141] | V. Lakshmikantham, <em>Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems</em>,, Mathematics and its Applications (1991) · Zbl 0721.34054 · 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, 5936 (2010) · doi:10.1109/CDC.2010.5717266 |
| [143] | M. Lazar, On infinity norms as Lyapunov functions for continuous-time dynamical systems,, in Proceedings of the 50th IEEE Conference on Decision and Control, 7567 (2011) · doi:10.1109/CDC.2011.6161163 |
| [144] | M. Lazar, On stability analysis of discrete-time homogeneous dynamics,, in Proceedings of the 17th International Conference on Systems Theory, 297 (2013) · doi:10.1109/ICSTCC.2013.6688976 |
| [145] | M. Lazar, On infinity norms as Lyapunov functions for piecewise affine systems,, in HSCC’10, 131 (2010) · Zbl 1360.93554 · doi:10.1145/1755952.1755972 |
| [146] | G. Leonov, <em>Frequency Methods in Oscillation Theory</em>,, Mathematics and its Applications (1996) · Zbl 0844.34005 · doi:10.1007/978-94-009-0193-3 |
| [147] | D. Lewis, Metric properties of differential equations,, Amer. J. Math., 71, 294 (1949) · Zbl 0034.39203 · doi:10.2307/2372245 |
| [148] | H. Li, 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), 1189 (0158) · Zbl 1332.93328 |
| [149] | H. Li, Computation of local ISS Lyapunov functions with low gains via linear programming,, Discrete Contin. Dyn. Syst. Ser. B, 8, 2477 (2015) · Zbl 1366.37146 · doi:10.3934/dcdsb.2015.20.2477 |
| [150] | H. Li, 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, 5512 (2014) · doi:10.1109/CDC.2014.7040251 |
| [151] | D. Liberzon, <em>Switching in Systems and Control</em>,, Systems & Control: Foundations & Applications (2003) · Zbl 1036.93001 · doi:10.1007/978-1-4612-0017-8 |
| [152] | Y. Lin, A smooth converse Lyapunov theorem for robust stability,, SIAM J. Control Optimization, 34, 124 (1996) · Zbl 0856.93070 · doi:10.1137/S0363012993259981 |
| [153] | Z. Liu, The random case of Conley’s theorem,, Nonlinearity, 19, 277 (2006) · Zbl 1134.37349 · 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, 1017 (2007) · Zbl 1124.37031 · 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, 2773 (2007) · Zbl 1138.37031 · doi:10.1088/0951-7715/20/12/003 |
| [156] | W. Lohmiller, On Contraction Analysis for Non-linear Systems,, Automatica, 34, 683 (1998) · Zbl 0934.93034 · doi:10.1016/S0005-1098(98)00019-3 |
| [157] | K. Loparo, Estimating the domain of attraction of nonlinear feedback systems,, IEEE Trans. Automat. Control, 23, 602 (1978) · Zbl 0385.93023 |
| [158] | A. Lyapunov, The general problem of the stability of motion,, Internat. J. Control, 55, 521 (1992) · Zbl 0786.70001 · doi:10.1080/00207179208934253 |
| [159] | M. Malisoff, <em>Constructions of Strict Lyapunov Functions</em>,, Communications and Control Engineering (2009) · Zbl 1186.93001 · doi:10.1007/978-1-84882-535-2 |
| [160] | I. Manchester, Transverse contraction criteria for existence, stability, and robustness of a limit cycle,, Systems Control Lett., 63, 32 (2014) · Zbl 1283.93142 · doi:10.1016/j.sysconle.2013.10.005 |
| [161] | X. Mao, <em>Stochastic Differential Equations and Applications</em>,, 2nd edition (2008) · doi:10.1533/9780857099402 |
| [162] | S. Margolis, Control engineering applications of V. I. Zubov’s construction procedure for Lyapunov functions,, IEEE Trans. Automat. Control, 8, 104 (1963) · 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, 137 (2002) · Zbl 1013.34051 · doi:10.1080/0268111011011847 |
| [164] | S. Marinósson, <em>Stability Analysis of Nonlinear Systems with Linear Programming: A Lyapunov Functions Based Approach</em>,, PhD thesis (2002) · Zbl 1344.37001 |
| [165] | A. Martynyuk, Analysis of stability problems via Matrix Lyapunov Functions,, J. Appl. Math. Stochastic Anal., 3, 209 (1990) · Zbl 0714.34082 · doi:10.1155/S104895339000020X |
| [166] | J. Massera, On Liapounoff’s conditions of stability,, Ann. of Math., 50, 705 (1949) · Zbl 0038.25003 · doi:10.2307/1969558 |
| [167] | J. Massera, Contributions to stability theory,, Ann. of Math., 64, 182 (1956) · Zbl 0070.31003 · doi:10.2307/1969955 |
| [168] | V. Matrosov, On the stability of motion,, J. Appl. Math. Mech., 26, 1337 (1963) · Zbl 0123.05404 |
| [169] | P. Menck, How basin stability complements the linear-stability paradigm,, Nature Physics, 9, 89 (2013) · doi:10.1038/nphys2516 |
| [170] | S. Meyn, <em>Markov Chains and Stochastic Stability</em>,, 2nd edition (2009) · Zbl 1165.60001 · doi:10.1017/CBO9780511626630 |
| [171] | A. Michel, <em>Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems</em>,, Systems & Control: Foundations & Applications (2008) · Zbl 1146.34002 |
| [172] | A. Michel, Stability analysis of interconnected systems using computer generated Lyapunov functions,, IEEE Trans. Circuits and Systems, 29, 431 (1982) · Zbl 0492.93049 · doi:10.1109/TCS.1982.1085181 |
| [173] | A. Michel, Computer generated Lyapunov functions for interconnected systems: Improved results with applications to power system,, IEEE Trans. Circuits and Systems, 31, 189 (1984) · Zbl 0542.93050 · doi:10.1109/TCS.1984.1085483 |
| [174] | A. Michel, Stability analysis of complex dynamical systems,, Circuits Systems Signal Process, 1, 171 (1982) · Zbl 0493.93040 · doi:10.1007/BF01600051 |
| [175] | C. Mikkelsen, <em>Numerical Methods For Large Lyapunov Equations</em>,, PhD thesis (2009) |
| [176] | N. Mohammed, Grid refinement in the construction of Lyapunov functions using radial basis functions,, Discrete Contin. Dyn. Syst. Ser. B, 8, 2453 (2015) · Zbl 1366.37147 · doi:10.3934/dcdsb.2015.20.2453 |
| [177] | A. Molchanov, Criteria of asymptotic stability of differential and difference inclusions encountered in control theory,, Systems Control Lett., 13, 59 (1989) · Zbl 0684.93065 · doi:10.1016/0167-6911(89)90021-2 |
| [178] | A. Molchanov, Lyapunov functions that specify necessary and sufficient conditions of absolute stability of nonlinear nonstationary control systems I, II,, Automat. Remote Control, 47, 344 (1986) · Zbl 0607.93039 |
| [179] | N. Noroozi, 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, 585 (1995) · Zbl 0847.58049 |
| [181] | Y. Ohta, On the construction of piecewise linear Lyapunov functions,, in Proceedings of the 40th IEEE Conference on Decision and Control, 3, 2173 (2001) · doi:10.1109/CDC.2001.980577 |
| [182] | Y. Ohta, A generalization of piecewise linear Lyapunov functions,, in Proceedings of the 42nd IEEE Conference on Decision and Control, 5, 5091 (2003) · 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, 62 (1964) |
| [184] | S. Panikhom, Numerical approach to construction of Lyapunov function for nonlinear stability analysis,, Research Journal of Applied Sciences, 4, 2915 (2012) |
| [185] | A. Papachristodoulou, <em>SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB</em>,, User’s Guide (2013) |
| [186] | P. Parrilo, <em>Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiziation</em>,, PhD thesis (2000) |
| [187] | M. Patrão, Existence of complete Lyapunov functions for semiflows on separable metric spaces,, Far East J. Dyn. Syst., 17, 49 (2011) · Zbl 1236.37015 |
| [188] | M. Peet, Exponentially stable nonlinear systems have polynomial Lyapunov functions on bounded regions,, IEEE Trans. Automat. Control, 54, 979 (2009) · Zbl 1367.93442 · doi:10.1109/TAC.2009.2017116 |
| [189] | M. Peet, A converse sum of squares Lyapunov result with a degree bound,, IEEE Trans. Automat. Control, 57, 2281 (2012) · Zbl 1369.93449 · doi:10.1109/TAC.2012.2190163 |
| [190] | S. Pettersson, Stability and robustness for hybrid systems,, in Proceedings of the 35th IEEE Conference on Decision and Control, 1202 (1996) · doi:10.1109/CDC.1996.572653 |
| [191] | A. Polanski, Lyapunov functions construction by linear programming,, IEEE Trans. Automat. Control, 42, 1113 (1997) · Zbl 0885.93050 · doi:10.1109/9.599986 |
| [192] | A. Polanski, On absolute stability analysis by polyhedral Lyapunov functions,, Automatica, 36, 573 (2000) · Zbl 0980.93073 · doi:10.1016/S0005-1098(99)00180-6 |
| [193] | I. Pólik, A survey of the S-lemma,, SIAM Review, 49, 371 (2007) · Zbl 1128.90046 · doi:10.1137/S003614450444614X |
| [194] | C. Prieur, ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws,, Math. Control Signals Syst., 24, 111 (2012) · Zbl 1238.93089 · 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, 1028 (1994) · doi:10.1109/ICNN.1994.374324 |
| [196] | S. Raković, The Minkowski-Lyapunov equation for linear dynamics: Theoretical foundations,, Automatica, 50, 2015 (2014) · Zbl 1297.93124 · doi:10.1016/j.automatica.2014.05.023 |
| [197] | M. Rasmussen, <em>Attractivity and Bifurcation for Nonautonomous Dynamical Systems</em>,, Lecture Notes in Mathematics (1907) · Zbl 1131.37001 |
| [198] | M. Rasmussen, Morse decompositions of nonautonomous dynamical systems,, Trans. Amer. Math. Soc., 359, 5091 (2007) · Zbl 1116.37013 · doi:10.1090/S0002-9947-07-04318-8 |
| [199] | S. Ratschan, Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions,, SIAM J. Control Optim., 48, 4377 (2010) · Zbl 1215.65188 · doi:10.1137/090749955 |
| [200] | M. Rezaiee-Pajand, Estimating the region of attraction via collocation for autonomous nonlinear systems,, Structural Engineering and Mechanics, 41, 263 (2012) · Zbl 1278.74185 |
| [201] | M. Roozbehani, Optimization of Lyapunov invariants in verification of software systems,, IEEE Trans. Automat. Control, 58, 696 (2013) · Zbl 1369.68155 · doi:10.1109/TAC.2013.2241472 |
| [202] | B. Rüffer, Convergent systems vs. incremental stability,, Systems Control Lett., 62, 277 (2013) · Zbl 1261.93072 · 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, 735 (2005) · doi:10.1109/IJCNN.2005.1555943 |
| [204] | Z. She, Discovering polynomial Lyapunov functions for continuous dynamical systems,, J. Symbolic Comput., 58, 41 (2013) · Zbl 1338.37025 · doi:10.1016/j.jsc.2013.06.003 |
| [205] | Z. She, Computing an invariance kernel with target by computing Lyapunov-like functions,, IET Control Theory Appl., 7, 1932 (2013) · doi:10.1049/iet-cta.2013.0275 |
| [206] | R. Shorten, Stability criteria for switched and hybrid systems,, SIAM Review, 49, 545 (2007) · Zbl 1127.93005 · doi:10.1137/05063516X |
| [207] | D. Šiljak, <em>Large-scale Dynamic Systems. Stability and Structure</em>,, North-Holland Series in System Science and Engineering (1979) · Zbl 1243.93001 |
| [208] | E. Sontag, A Lyapunov-like characterization of asymptotic controllability,, SIAM J. Control Optimization, 21, 462 (1983) · Zbl 0513.93047 · doi:10.1137/0321028 |
| [209] | E. Sontag, Smooth stabilization implies coprime factorization,, IEEE Trans. Automat. Control, 34, 435 (1989) · Zbl 0682.93045 · doi:10.1109/9.28018 |
| [210] | E. Sontag, New characterizations of input-to-state stability,, IEEE Trans. Automat. Control, 41, 1283 (1996) · Zbl 0862.93051 · doi:10.1109/9.536498 |
| [211] | E. Sontag, <em>Mathematical Control Theory</em>,, 2nd edition (1998) · Zbl 0945.93001 · doi:10.1007/978-1-4612-0577-7 |
| [212] | E. Sontag, Nonsmooth control-Lyapunov functions,, in Proceedings of the 34th IEEE Conference on Decision and Control, 2799 (1995) · doi:10.1109/CDC.1995.478542 |
| [213] | E. Sontag, On characterizations of the input-to-state stability property,, Systems Control Lett., 24, 351 (1995) · Zbl 0877.93121 · doi:10.1016/0167-6911(94)00050-6 |
| [214] | B. Stenström, Dynamical systems with a certain local contraction property,, Math. Scand., 11, 151 (1962) · Zbl 0178.24803 |
| [215] | A. Subbaraman, A converse Lyapunov theorem for strong global recurrence,, Automatica, 49, 2963 (2013) · Zbl 1358.93163 · doi:10.1016/j.automatica.2013.07.001 |
| [216] | A. Subbaraman, A Matrosov theorem for strong global recurrence,, Automatica, 49, 3390 (2013) · Zbl 1315.93074 · doi:10.1016/j.automatica.2013.08.009 |
| [217] | Z. Sun, Stability of piecewise linear systems revisited,, Annu. Rev. Control, 34, 221 (2010) · doi:10.1016/j.arcontrol.2010.08.003 |
| [218] | Z. Sun, <em>Stability Theory of Switched Dynamical Systems</em>,, Communications and Control Engineering (2011) · Zbl 1298.93006 · doi:10.1007/978-0-85729-256-8 |
| [219] | K. Tanaka, A multiple Lyapunov function approach to stabilization of fuzzy control systems,, IEEE T. Fuzzy Syst., 11, 582 (2003) · doi:10.1109/TFUZZ.2003.814861 |
| [220] | A. R. Teel, A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions,, ESAIM Control Optim. Calc. Var., 5, 313 (2000) · Zbl 0953.34042 · doi:10.1051/cocv:2000113 |
| [221] | R. Temam, <em>Infinite-Dimensional Dynamical Systems in Mechanics and Physics</em>,, Applied Mathematical Sciences (1997) · Zbl 0871.35001 · doi:10.1007/978-1-4612-0645-3 |
| [222] | A. Tesi, 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, 1907 (1994) · doi:10.1109/CDC.1994.411100 |
| [223] | A. Vannelli, Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems,, Automatica, 21, 69 (1985) · Zbl 0559.34052 · doi:10.1016/0005-1098(85)90099-8 |
| [224] | K. Wang, On the stability of a family of nonlinear time-varying system,, IEEE Trans. Circuits and Systems, 43, 517 (1996) · doi:10.1109/81.508171 |
| [225] | H. Wendland, <em>Scattered Data Approximation</em>,, Cambridge Monographs on Applied and Computational Mathematics (2005) · Zbl 1075.65021 |
| [226] | C. Xu, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems,, ESAIM Control Optim. Calc. Var., 7, 421 (2002) · Zbl 1040.93031 · doi:10.1051/cocv:2002062 |
| [227] | C. Yfoulis, A numerical technique for the stability analysis of linear switched systems,, Int. J. Control, 77, 1019 (2004) · Zbl 1069.93034 · doi:10.1080/002071704200026963 |
| [228] | T. Yoshizawa, <em>Stability Theory by Liapunov’s Second Method</em>,, Publications of the Mathematical Society of Japan (1966) · Zbl 0144.10802 |
| [229] | V. Zubov, <em>Methods of A. M. Lyapunov and Their Application</em>,, Translation prepared under the auspices of the United States Atomic Energy Commission; edited by Leo F. Boron (1964) · Zbl 0115.30204 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
