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Efficient computation of Lyapunov functions for Morse decompositions
Grid refinement in the construction of Lyapunov functions using radial basis functions
1. | Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom |
References:
[1] |
M. Berg, O. Cheong, M. Kerveld and M. Overmars, Computational Geometry: Algorithms and Applications, Springer-Verlag, Berlin, 2008. |
[2] |
M. D. Buhmann, Radial basis functions, in Acta Numerica, 2000, Acta Numer., 9, Cambridge Univ. Press, Cambridge, 2000, 1-38.
doi: 10.1017/S0962492900000015. |
[3] |
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. |
[4] |
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. |
[5] |
M. Floater and A. Iske, Multistep scattered data interpolation using compactly supported Radial Basis Functions, J. Comput. Appl. Math., 73 (1996), 65-78.
doi: 10.1016/0377-0427(96)00035-0. |
[6] |
P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math., 1904, Springer, 2007. |
[7] |
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. |
[8] |
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. |
[9] |
P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2291-2331.
doi: 10.3934/dcdsb.2015.20.2291. |
[10] |
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. |
[11] |
L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337.
doi: 10.1007/s002110050241. |
[12] |
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. |
[13] |
S. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete and Continuous Dynamical Systems - Series A, 10 (2004), 657-678.
doi: 10.3934/dcds.2004.10.657. |
[14] |
S. Hafstein, An algorithm for constructing Lyapunov functions, Monograph. Electron. J. Diff. Eqns., (2007), 101pp. |
[15] |
C. S. Hsu, Cell-to-cell Mapping, Applied Mathematical Sciences, 64, Springer-Verlag, New York, 1987.
doi: 10.1007/978-1-4757-3892-6. |
[16] |
A. Iske, On the construction of kernel-based adaptive particle methods in numerical flow simulation, in Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, Notes Numer. Fluid Mech. Multidiscip. Des., 120, Springer, Heidelberg, 2013, 197-221.
doi: 10.1007/978-3-642-33221-0_12. |
[17] |
S. Iyengar, K. Boroojeni and N. Balakrishnan, Mathematical Theories of Distributed Sensor Networks, Springer, New York, 2014.
doi: 10.1007/978-1-4419-8420-3. |
[18] |
Z. Jian, Development of Strong Form Methods with Applications in Computational Mechanics, PhD thesis, National University of Singapore, Singapore, 2008. |
[19] |
C. Kellett, Classical converse theorems in Lyapunov’s second method, Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2333-2360.
doi: 10.3934/dcdsb.2015.20.2333. |
[20] |
R. Klein, Concrete and Abstract Voronoi Diagrams, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1989.
doi: 10.1007/3-540-52055-4. |
[21] |
J. Massera, On Liapounoff's conditions of stability, Ann. of Math., 50 (1949), 705-721. |
[22] |
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. |
[23] |
P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiza, PhD thesis, California Institute of Technology Pasadena, California, 2000. |
[24] |
F. Preparata and M. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985.
doi: 10.1007/978-1-4612-1098-6. |
[25] |
J. Ruppert, A Delaunay refinement algorithm for quality 2-dimensional mesh generation, J. Approx. Theory, 18 (1995), 548-585.
doi: 10.1006/jagm.1995.1021. |
[26] |
R. Sibson, Development of strong form methods with applications in computational mechanics, in Interpolating Multivariate Data, Chapter 2 (ed. V. Barnett), John Wiley and Sons, New York, 1981. |
[27] |
H. Wendland, Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272.
doi: 10.1006/jath.1997.3137. |
[28] |
H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005. |
[29] |
X. Zhang, R. Ding and Y. Li, Adaptive RPIM meshless method, in Proceedings of the 2011 International Conference on Multimedia Technology (ICMT), IEEE, 2011, 2388-2392. |
show all references
References:
[1] |
M. Berg, O. Cheong, M. Kerveld and M. Overmars, Computational Geometry: Algorithms and Applications, Springer-Verlag, Berlin, 2008. |
[2] |
M. D. Buhmann, Radial basis functions, in Acta Numerica, 2000, Acta Numer., 9, Cambridge Univ. Press, Cambridge, 2000, 1-38.
doi: 10.1017/S0962492900000015. |
[3] |
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. |
[4] |
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. |
[5] |
M. Floater and A. Iske, Multistep scattered data interpolation using compactly supported Radial Basis Functions, J. Comput. Appl. Math., 73 (1996), 65-78.
doi: 10.1016/0377-0427(96)00035-0. |
[6] |
P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math., 1904, Springer, 2007. |
[7] |
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. |
[8] |
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. |
[9] |
P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2291-2331.
doi: 10.3934/dcdsb.2015.20.2291. |
[10] |
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. |
[11] |
L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337.
doi: 10.1007/s002110050241. |
[12] |
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. |
[13] |
S. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete and Continuous Dynamical Systems - Series A, 10 (2004), 657-678.
doi: 10.3934/dcds.2004.10.657. |
[14] |
S. Hafstein, An algorithm for constructing Lyapunov functions, Monograph. Electron. J. Diff. Eqns., (2007), 101pp. |
[15] |
C. S. Hsu, Cell-to-cell Mapping, Applied Mathematical Sciences, 64, Springer-Verlag, New York, 1987.
doi: 10.1007/978-1-4757-3892-6. |
[16] |
A. Iske, On the construction of kernel-based adaptive particle methods in numerical flow simulation, in Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, Notes Numer. Fluid Mech. Multidiscip. Des., 120, Springer, Heidelberg, 2013, 197-221.
doi: 10.1007/978-3-642-33221-0_12. |
[17] |
S. Iyengar, K. Boroojeni and N. Balakrishnan, Mathematical Theories of Distributed Sensor Networks, Springer, New York, 2014.
doi: 10.1007/978-1-4419-8420-3. |
[18] |
Z. Jian, Development of Strong Form Methods with Applications in Computational Mechanics, PhD thesis, National University of Singapore, Singapore, 2008. |
[19] |
C. Kellett, Classical converse theorems in Lyapunov’s second method, Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2333-2360.
doi: 10.3934/dcdsb.2015.20.2333. |
[20] |
R. Klein, Concrete and Abstract Voronoi Diagrams, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1989.
doi: 10.1007/3-540-52055-4. |
[21] |
J. Massera, On Liapounoff's conditions of stability, Ann. of Math., 50 (1949), 705-721. |
[22] |
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. |
[23] |
P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiza, PhD thesis, California Institute of Technology Pasadena, California, 2000. |
[24] |
F. Preparata and M. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985.
doi: 10.1007/978-1-4612-1098-6. |
[25] |
J. Ruppert, A Delaunay refinement algorithm for quality 2-dimensional mesh generation, J. Approx. Theory, 18 (1995), 548-585.
doi: 10.1006/jagm.1995.1021. |
[26] |
R. Sibson, Development of strong form methods with applications in computational mechanics, in Interpolating Multivariate Data, Chapter 2 (ed. V. Barnett), John Wiley and Sons, New York, 1981. |
[27] |
H. Wendland, Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272.
doi: 10.1006/jath.1997.3137. |
[28] |
H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005. |
[29] |
X. Zhang, R. Ding and Y. Li, Adaptive RPIM meshless method, in Proceedings of the 2011 International Conference on Multimedia Technology (ICMT), IEEE, 2011, 2388-2392. |
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