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Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary
Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function
1. | Faculty of Physical Sciences, University of Iceland, 107 Reykjavik, Iceland |
2. | Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom |
3. | Svensk Exportkredit, Klarabergsviadukten 61-63, 111 64 Stockholm, Sweden |
The γ-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic γ-basin of attraction of the equilibrium.
References:
[1] |
H. Björnsson, P. Giesl, S. Gudmundsson and S. Hafstein, Local Lyapunov functions for nonlinear stochastic differential equations by linearization, In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, 2018,579–586, . |
[2] |
M. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511543241. |
[3] |
F. Camilli and L. Grüne,
Characterizing attraction probabilities via the stochastic Zubov equation, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 457-468.
doi: 10.3934/dcdsb.2003.3.457. |
[4] |
P. Giesl, Construction of Global Lyapunov functions using Radial Basis Functions, volume 1904 of Lecture Notes in Mathematics, Springer, Berlin, 2007. |
[5] |
P. Giesl and S. Hafstein,
Review of computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331.
doi: 10.3934/dcdsb.2015.20.2291. |
[6] |
P. Giesl and N. Mohammed, Verification estimates for the construction of Lyapunov functions using meshfree collocation, Discrete Contin. Dyn. Syst. Ser. B, in press. |
[7] |
P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Numer. Anal., 45 92007), 1723–1741.
doi: 10.1137/060658813. |
[8] |
S. Gudmundsson and S. Hafstein, Probabilistic basin of attraction and its estimation using two Lyapunov functions, Complexity, 2018 (2018), Article ID 2895658, 9 pages.
doi: 10.1155/2018/2895658. |
[9] |
S. Hafstein, S. Gudmundsson, P. Giesl and E. Scalas,
Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 2 (2018), 939-956.
doi: 10.3934/dcdsb.2018049. |
[10] |
N. Mohammed, Grid Refinement and Verification Estimates for the RBF Construction Method of Lyapunov Functions, PhD thesis, University of Sussex, 2016. |
[11] |
M. J. D. Powell, The theory of radial basis function approximation in 1990, In Advances in Numerical Analysis, Vol. Ⅱ (Lancaster, 1990), Oxford Sci. Publ., pages 105-210. Oxford Univ. Press, New York, 1992. |
[12] |
R. Schaback and H. Wendland,
Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639.
doi: 10.1017/S0962492906270016. |
[13] |
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. |
[14] |
H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005. |
show all references
References:
[1] |
H. Björnsson, P. Giesl, S. Gudmundsson and S. Hafstein, Local Lyapunov functions for nonlinear stochastic differential equations by linearization, In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, 2018,579–586, . |
[2] |
M. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511543241. |
[3] |
F. Camilli and L. Grüne,
Characterizing attraction probabilities via the stochastic Zubov equation, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 457-468.
doi: 10.3934/dcdsb.2003.3.457. |
[4] |
P. Giesl, Construction of Global Lyapunov functions using Radial Basis Functions, volume 1904 of Lecture Notes in Mathematics, Springer, Berlin, 2007. |
[5] |
P. Giesl and S. Hafstein,
Review of computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331.
doi: 10.3934/dcdsb.2015.20.2291. |
[6] |
P. Giesl and N. Mohammed, Verification estimates for the construction of Lyapunov functions using meshfree collocation, Discrete Contin. Dyn. Syst. Ser. B, in press. |
[7] |
P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Numer. Anal., 45 92007), 1723–1741.
doi: 10.1137/060658813. |
[8] |
S. Gudmundsson and S. Hafstein, Probabilistic basin of attraction and its estimation using two Lyapunov functions, Complexity, 2018 (2018), Article ID 2895658, 9 pages.
doi: 10.1155/2018/2895658. |
[9] |
S. Hafstein, S. Gudmundsson, P. Giesl and E. Scalas,
Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 2 (2018), 939-956.
doi: 10.3934/dcdsb.2018049. |
[10] |
N. Mohammed, Grid Refinement and Verification Estimates for the RBF Construction Method of Lyapunov Functions, PhD thesis, University of Sussex, 2016. |
[11] |
M. J. D. Powell, The theory of radial basis function approximation in 1990, In Advances in Numerical Analysis, Vol. Ⅱ (Lancaster, 1990), Oxford Sci. Publ., pages 105-210. Oxford Univ. Press, New York, 1992. |
[12] |
R. Schaback and H. Wendland,
Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639.
doi: 10.1017/S0962492906270016. |
[13] |
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. |
[14] |
H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005. |

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