Construction of global lyapunov functions using radial basis functions

Lecture Notes in Mathematics

Volumn 1904, Issue , 2007, Pages 1-172

  Giesl, Peter ^a  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

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EID: 34247100563     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: None     Document Type: Article

References (71)

1. R. ABRAHAM & J. E. MARSDEN, Foundations of Mechanics, Addison-Wesley, Redwood City, CA, 1985 (2nd ed.).
2. H. AMANN, Ordinary Differential Equations. An Introduction to Nonlinear Analysis, de Gruyter, Studies in Mathematics 13, Berlin, 1990.
3. B. AULBACH, Asymptotic Stability regions via Extensions of Zubov's Method - I, Nonlinear Anal. 7 (1983) 12, 1431-1440.
4. B. AULBACH, Asymptotic Stability regions via Extensions of Zubov's Method - II, Nonlinear Anal. 7 (1983) 12, 1441-1454.
5. B. AULBACH, Gewöhnliche Differenzialgleichungen, Spektrum Akademischer Verlag, Heidelberg, 2004 (2nd ed.).
6. E. A. BARBAŠIN & N. N. KRASOVSKIǏ, On the existence of Liapunov functions in the case of asymptotic stability in the large, Prikl. Mat. Mekh. 18 (1954), 345-350.
7. A. BARREIRO, J. ARACIL & D. PAGANO, Detection of attraction domains of non-linear systems using bifurcation analysis and Lyapunov functions, Internat. J. Control 75 (2002), 314-327.
8. N. BHATIA, On asymptotic stability in dynamical systems, Math. Systems Theory 1 (1967), 113-128.
9. N. BHATIA & G. SZEGÖ, Stability Theory of Dynamical Systems, Grundlehren der mathematischen Wissenschaften 161, Springer, Berlin, 1970.
10. M. D. BUHMANN, Radial basis functions, Acta Numer. 9 (2000), 1-38.
11. M. D. BUHMANN, Radial basis functions: theory and implementations, Cambridge University Press, Cambridge, 2003.
12. F. CAMILLI, L. GRÜNE & F. WIRTH, A regularization of Zubov's equation for robust domains of attraction, in: Nonlinear Control in the Year 2000, A. Isidori et al. (eds.), Lecture Notes in Control and Information Sciences 258, Springer, London, 2000, 277-290.
13. F. CAMILLI, L. GRÜNE & F. WIRTH, A generalization of Zubov's method to perturbed systems, SIAM J. Control Optim. 40 (2001) 2, 496-515.
14. H.-D. CHIANG, M. W. HIRSCH & F. F. WU, Stability Regions of Nonlinear Autonomous Dynamical Systems, IEEE Trans. Automat. Control 33 (1988) 1, 16-27.
15. F. H. CLARKE, Y. S. LEDAYEV & R. J. STERN, Asymptotic stability and smooth Lyapunov functions, J. Differential Equations 149 (1998), 69-115.
16. F. FALLSIDE & M. R. PATEL, Step-response behaviour of a speed-control system with a back-e.m.f. nonlineanty, Proc. IEE (London) 112 (1965) 10, 1979-1984.
17. C. FRANKE & R. SCHABACK, Convergence order estimates of meshless collocation methods using radial basis functions, Adv. Comput. Math. 8 (1998), 381-399.
18. C. FRANKE & R. SCHABACK, Solving Partial Differential Equations by Collocation using Radial Basis Functions, Appl. Math. Comput. 93 (1998) 1, 73-82.
19. R. GENESIO, M. TARTAGLIA & A. VICINO, On the Estimation of Asymptotic Stability Regions: State of the Art and New Proposals, IEEE Trans. Automat. Control 30 (1985), 747-755.
20. P. GIESL, Necessary Conditions for a Limit Cycle and its Basin of Attraction, Nonlinear Anal. 56 (2004) 5, 643-677.
21. P. GIESL, Construction of global Lyapunov functions using radial basis functions, Habilitation Thesis, TU München, 2004.
22. P. GIESL, Approximation of domains of attraction and Lyapunov functions using radial basis functions, in: Proceedings of the NOLCOS 2004 Conference in Stuttgart, Germany, Vol. 2, 2004, 865-870.
23. P. GIESL, Stepwise calculation of the basin of attraction in dynamical systems using radial basis functions, in: "Algorithms for Approximation", A. Iske & J. Levesley (eds.), Springer, Heidelberg, 2007, 113-122.
24. P. GIESL, Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete Cont. Dyn. Sys. Ser. B 7 (2007) No. 1, 101-124.
25. P. GIESL, On the determination of the basin of attraction of discrete dynamical systems, to appear in J. Difference Equ. Appl.
26. P. GIESL, On the determination of the basin of attraction of a periodic orbit in two-dimensional systems. Preprint, Munich, 2006.
27. P. GIESL & H. WENDLAND, Meshless Collocation: Error estimates with Application to Dynamical Systems (submitted to SIAM J. Numer. Anal.)
28. P. GIESL & H. WENDLAND, Approximating the basin of attraction of time-periodic ODEs by meshless collocation. Preprint Göttingen/Munich, 2006.
29. P. GLENDINNING, Stability, instability and chaos: an introduction to the theory of nonlinear differential equations, Cambridge University Press, Cambridge, 1994.
30. L. GRÜNE, An adaptive grid scheme for the discrete Hamitton-Jacobi-Bellman equation, Numer. Math. 75 (1997), 319-337.
31. L. GRÜNE, E. D. SONTAG & F. WIRTH, Asymptotic stability equals exponential stability, and ISS equals finite energy gain - if you twist your eyes, Systems Control Lett. 38 (1999) 2, 127-134.
32. S. HAFSTEIN, A constructive converse Lyapunov Theorem on Exponential Stability, Discrete Contin. Dyn. Syst. 10 (2004) 3, 657-678.
33. W. HAHN, Eine Bemerkung zur zweiten Methode von Ljapunov, Math. Nachr. 14 (1956), 349-354.
34. W. HAHN, Theorie und Anwendung der direkten Methode von Ljapunov, Ergebnisse der Mathematik und ihrer Grenzgebiete 22, Springer, Berlin, 1959.
35. W. HAHN, Stability of Motion, Springer, New York, 1967.
36. PH. HARTMAN, Ordinary Differential Equations, Wiley, New York, 1964.
37. L. HÖRMANDER, Linear Partial Differential Operators, Springer, Berlin, 1963.
38. A. ISKE, Reconstruction of Functions from Generalized Hermite-Birkhoff Data, in: Approximation Theory VIII, Vol. 1: Approximation and Interpolation, Ch. Chui & L. Schumaker (eds.), 1995, 257-264.
39. A. ISKE, Perfect Centre Placement for Radial Basis Function Methods, Technical Report, TUM-M9809, TU München, 1998.
40. A. ISKE, Scattered Data Modelling Using Radial Basis Functions, in: Tutorials on Multiresolution in Geometric Modelling, A. Iske, E. Quak & M. Floater (eds.), Springer, 2002, 205-242.
41. L. B. JOCIĆ, Planar regions of attraction, IEEE Trans. Automat. Control 27 (1982) 3, 708-710.
42. P. JULIÁN, J. GUIVANT & A. DESAGES, A parametrization of piecewise linear Lyapunov functions via linear programming, Int. J. Control 72 (1999), 702-715.
43. E. J. KANSA, Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics, i: Surface approximations and partial derivative, estimates, Comput. Math. Appl. 19 (1990), 127-145.
44. E. J. KANSA, Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics, ii: Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl. 19 (1990), 147-161.
45. N. E. KIRIN, R. A. NELEPIN & V. N. BAIDAEV, Construction of the Attraction Region by Zubov's method, Differ. Equ. 17 (1981), 871-880.
46. N. N. KRASOVSKIǏ, Stability of Motion, Stanford University Press, Stanford, 1963. Translation of the russian original, Moscow, 1959.
47. Y. LIN, E. D. SONTAG & Y. WANG, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim. 34 (1996), 124-160.
48. A. M. LYAPUNOV, Problème general de la stabilité du mouvement, Ann. Fac. Sci. Toulouse 9 (1907), 203-474.
49. Translation of the russian version, published 1893 in Comm. Soc. math. Kharkow. Newly printed: Ann. of math. Stud. 17, Princeton, 1949.
50. S. MARINOSSÓN, Stability Analysis of Nonlinear Systems with Linear Programming. A Lyapunov Functions Based Approach, Ph.D. thesis, University of Duisburg, Germany, 2002.
51. J. L. MASSERA, On Liapounoff's Conditions of Stability, Ann. of Math. 50 (1949) 2, 705-721.
52. F. J. NARCOWICH, J. D. WARD & H. WENDLAND, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comp. 74 (2005), 643-763.
53. M. J. D. POWELL, The Theory of Radial Basis Function Approximation in 1990, in: Advances in Numerical Analysis II: Wavelets, Subdivision Algorithms, and Radial Basis Functions, W. A. Light (ed.), Oxford University Press, 1992, 105-210.
54. R. SCHABACK, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math. 3 (1995), 251-264.
55. R. SCHABACK & H. WENDLAND, Using compactly supported radial basis functions to solve partial differential equations, in: Boundary Element Technology XIII (invited lecture), C. Chen, C. Brebbia & D. Pepper (eds.), WitPress, Southampton, Boston, 1999, 311-324.
56. H. R. SCHWARZ, Numerische Mathematik, Teubner, Stuttgart, 1988.
57. H. L. SMITH & P. WALTMAN, The theory of the chemostat, Cambridge University Press, 1995.
58. E. D. SONTAG, Mathematical control theory, Springer, 1998 (2nd ed.).
59. A. R. TEEL & 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.
60. A. VANNELLI & M. VIDYASAGAR, Maximal Lyapunov Functions and Domains of Attraction for Autonomous Nonlinear Systems, Automatica J. IFAC 21 (1985) 1, 69-80.
61. F. VERHULST, Nonlinear Differential Equations and Dynamical Systems, Springer, 1996.
62. H. WENDLAND, Konstruktion und Untersuchung radialer Basisfunktionen mit kompaktem Träger, Ph.D. thesis, University of Göttingen, Germany, 1996.
63. H. WENDLAND, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998), 258-272.
64. H. WENDLAND, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004.
65. F. WESLEY WILSON, JR., The structure of the Level Surfaces of a Lyapunov Function, J. Differential Equations 3 (1967), 323-329.
66. F. WESLEY WILSON, JR., Smoothing Derivatives of Functions and Applications, Trans. Amer. Math. Soc. 139 (1969), 413-428.
67. S. WIGGINS, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, 1990.
68. Z. WU, Hermite-Birkhoff interpolation of scattered data by radial basis functions, Approx. Theory Appl. 8 (1995), 283-292.
69. Z. WU & R. SCHABACK, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), 13-27.
70. K. YOSIDA, Functional Analysis, Springer, Berlin, 1974 (4th ed.).
71. V. I. ZUBOV, Methods of A.M. Lyapunov and their Application, P. Noordhoff, Groningen, 1964.