-
Previous Article
Computation of local ISS Lyapunov functions with low gains via linear programming
- DCDS-B Home
- This Issue
-
Next Article
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, (2008). Google Scholar |
[2] |
M. D. Buhmann, Radial basis functions,, in Acta Numerica, (2000), 1.
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.
doi: 10.1137/S036301299936316X. |
[4] |
M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in Handbook of Dynamical Systems, (2002), 221.
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.
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).
|
[7] |
P. Giesl, Construction of a local and global Lyapunov function using Radial Basis Functions,, IMA J. Appl. Math., 73 (2008), 782.
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.
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.
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.
doi: 10.1137/060658813. |
[11] |
L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation,, Numer. Math., 75 (1997), 319.
doi: 10.1007/s002110050241. |
[12] |
L. Grüne, Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization,, Lecture Notes in Mathematics, (1783).
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.
doi: 10.3934/dcds.2004.10.657. |
[14] |
S. Hafstein, An algorithm for constructing Lyapunov functions,, Monograph. Electron. J. Diff. Eqns., (2007). Google Scholar |
[15] |
C. S. Hsu, Cell-to-cell Mapping,, Applied Mathematical Sciences, (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, (2013), 197.
doi: 10.1007/978-3-642-33221-0_12. |
[17] |
S. Iyengar, K. Boroojeni and N. Balakrishnan, Mathematical Theories of Distributed Sensor Networks,, Springer, (2014).
doi: 10.1007/978-1-4419-8420-3. |
[18] |
Z. Jian, Development of Strong Form Methods with Applications in Computational Mechanics,, PhD thesis, (2008). Google Scholar |
[19] |
C. Kellett, Classical converse theorems in Lyapunov’s second method,, Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2333.
doi: 10.3934/dcdsb.2015.20.2333. |
[20] |
R. Klein, Concrete and Abstract Voronoi Diagrams,, Lecture Notes in Computer Science, (1989).
doi: 10.1007/3-540-52055-4. |
[21] |
J. Massera, On Liapounoff's conditions of stability,, Ann. of Math., 50 (1949), 705.
|
[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). Google Scholar |
[23] |
P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiza,, PhD thesis, (2000). Google Scholar |
[24] |
F. Preparata and M. Shamos, Computational Geometry,, Texts and Monographs in Computer Science, (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.
doi: 10.1006/jagm.1995.1021. |
[26] |
R. Sibson, Development of strong form methods with applications in computational mechanics,, in Interpolating Multivariate Data, (1981). Google Scholar |
[27] |
H. Wendland, Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree,, J. Approx. Theory, 93 (1998), 258.
doi: 10.1006/jath.1997.3137. |
[28] |
H. Wendland, Scattered Data Approximation,, Cambridge Monographs on Applied and Computational Mathematics, (2005).
|
[29] |
X. Zhang, R. Ding and Y. Li, Adaptive RPIM meshless method,, in Proceedings of the 2011 International Conference on Multimedia Technology (ICMT), (2011), 2388. Google Scholar |
show all references
References:
[1] |
M. Berg, O. Cheong, M. Kerveld and M. Overmars, Computational Geometry: Algorithms and Applications,, Springer-Verlag, (2008). Google Scholar |
[2] |
M. D. Buhmann, Radial basis functions,, in Acta Numerica, (2000), 1.
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.
doi: 10.1137/S036301299936316X. |
[4] |
M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in Handbook of Dynamical Systems, (2002), 221.
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.
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).
|
[7] |
P. Giesl, Construction of a local and global Lyapunov function using Radial Basis Functions,, IMA J. Appl. Math., 73 (2008), 782.
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.
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.
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.
doi: 10.1137/060658813. |
[11] |
L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation,, Numer. Math., 75 (1997), 319.
doi: 10.1007/s002110050241. |
[12] |
L. Grüne, Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization,, Lecture Notes in Mathematics, (1783).
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.
doi: 10.3934/dcds.2004.10.657. |
[14] |
S. Hafstein, An algorithm for constructing Lyapunov functions,, Monograph. Electron. J. Diff. Eqns., (2007). Google Scholar |
[15] |
C. S. Hsu, Cell-to-cell Mapping,, Applied Mathematical Sciences, (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, (2013), 197.
doi: 10.1007/978-3-642-33221-0_12. |
[17] |
S. Iyengar, K. Boroojeni and N. Balakrishnan, Mathematical Theories of Distributed Sensor Networks,, Springer, (2014).
doi: 10.1007/978-1-4419-8420-3. |
[18] |
Z. Jian, Development of Strong Form Methods with Applications in Computational Mechanics,, PhD thesis, (2008). Google Scholar |
[19] |
C. Kellett, Classical converse theorems in Lyapunov’s second method,, Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2333.
doi: 10.3934/dcdsb.2015.20.2333. |
[20] |
R. Klein, Concrete and Abstract Voronoi Diagrams,, Lecture Notes in Computer Science, (1989).
doi: 10.1007/3-540-52055-4. |
[21] |
J. Massera, On Liapounoff's conditions of stability,, Ann. of Math., 50 (1949), 705.
|
[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). Google Scholar |
[23] |
P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiza,, PhD thesis, (2000). Google Scholar |
[24] |
F. Preparata and M. Shamos, Computational Geometry,, Texts and Monographs in Computer Science, (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.
doi: 10.1006/jagm.1995.1021. |
[26] |
R. Sibson, Development of strong form methods with applications in computational mechanics,, in Interpolating Multivariate Data, (1981). Google Scholar |
[27] |
H. Wendland, Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree,, J. Approx. Theory, 93 (1998), 258.
doi: 10.1006/jath.1997.3137. |
[28] |
H. Wendland, Scattered Data Approximation,, Cambridge Monographs on Applied and Computational Mathematics, (2005).
|
[29] |
X. Zhang, R. Ding and Y. Li, Adaptive RPIM meshless method,, in Proceedings of the 2011 International Conference on Multimedia Technology (ICMT), (2011), 2388. Google Scholar |
[1] |
Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709 |
[2] |
Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018 |
[3] |
Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 |
[4] |
Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021019 |
[5] |
Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021020 |
[6] |
Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810 |
[7] |
M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202 |
[8] |
Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003 |
[9] |
Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817 |
[10] |
Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 |
[11] |
Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109 |
[12] |
Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223 |
[13] |
Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 |
[14] |
Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069 |
[15] |
Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045 |
[16] |
Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]