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## 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

Received  April 2018 Revised  October 2018 Published  April 2019

Fund Project: The research for this paper was supported by the Icelandic Research Fund (Rannís) in the project `Lyapunov Methods and Stochastic Stability' (152429-051), which is gratefully acknowledged

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.

Citation: Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas, Skuli Gudmundsson. Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019080
##### 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.
Above: the computed non-local Lyapunov function $v$ for system (5.1). Below: the function $Lv$, approximating $-10^{-3}$
Non-local Lyapunov function for system (5.2) with $\theta = 1$. The non-local Lyapunov functions looks very similar to the one computed in [3]
The table shows the Wendland function $\psi_0(r): = \phi_{8, 6}(cr)$ as well as the related functions $\psi_1$ to $\psi_6$, defined recursively by $\psi_{k+1}(r): = \frac{\partial_r \psi_k(r)}{r}$ for $k = 0, 1, \ldots, 5$
 $\phi_{8, 6}$ $\psi_0(r)$ $[46,189(cr)^6+73,206 (cr)^5+54,915(cr)^4+24,500(cr)^3$ $+6,755(cr)^2+1,078cr+77]\, (1- cr)^{14}_+$ $\psi_1(r)$ $-380\, c^2\, [2,431(cr)^5+2,931(cr)^4+1,638(cr)^3+518(cr)^2$ $+91cr+7]\, (1- cr)^{13}_+$ $\psi_2(r)$ $12,920\, c^4\, [1,287(cr)^4+1,108(cr)^3+426(cr)^2$ $+84cr+7]\, (1- cr)^{12}_+$ $\psi_3(r)$ $-620,160\, c^6\, [429 (cr)^3+239 (cr)^2+55 cr+5]\, (1- cr)^{11}_+$ $\psi_4(r)$ $112,869,120\, c^8\, [33(cr)^2+10cr+1]\, (1- cr)_+^{10}$ $\psi_5(r)$ $-4,966,241,280\, c^{10}\, [9cr+1]\, (1- cr)_+^{9}$ $\psi_6(r)$ $446,961,715,200\, c^{12}\, (1- cr)_+^{8}$
 $\phi_{8, 6}$ $\psi_0(r)$ $[46,189(cr)^6+73,206 (cr)^5+54,915(cr)^4+24,500(cr)^3$ $+6,755(cr)^2+1,078cr+77]\, (1- cr)^{14}_+$ $\psi_1(r)$ $-380\, c^2\, [2,431(cr)^5+2,931(cr)^4+1,638(cr)^3+518(cr)^2$ $+91cr+7]\, (1- cr)^{13}_+$ $\psi_2(r)$ $12,920\, c^4\, [1,287(cr)^4+1,108(cr)^3+426(cr)^2$ $+84cr+7]\, (1- cr)^{12}_+$ $\psi_3(r)$ $-620,160\, c^6\, [429 (cr)^3+239 (cr)^2+55 cr+5]\, (1- cr)^{11}_+$ $\psi_4(r)$ $112,869,120\, c^8\, [33(cr)^2+10cr+1]\, (1- cr)_+^{10}$ $\psi_5(r)$ $-4,966,241,280\, c^{10}\, [9cr+1]\, (1- cr)_+^{9}$ $\psi_6(r)$ $446,961,715,200\, c^{12}\, (1- cr)_+^{8}$
The table shows the Wendland function $\psi_0(r): = \phi_{7, 6}(cr)$ as well as the related functions $\psi_1$ to $\psi_6$, defined recursively by $\psi_{k+1}(r): = \frac{\partial_r \psi_k(r)}{r}$ for $k = 0, 1, \ldots, 5$
 $\phi_{7, 6}$ $\psi_0(r)$ $[4,096(cr)^6+7,059 (cr)^5+5,751(cr)^4+2,782(cr)^3+830(cr)^2$ $+143cr+11]\, (1- cr)^{13}_+$ $\psi_1(r)$ $-38\, c^2\, [2,048(cr)^5+2,697(cr)^4+1,644(cr)^3+566(cr)^2$ $+108cr+9]\, (1- cr)^{12}_+$ $\psi_2(r)$ $10,336\, c^4\, [128(cr)^4+121(cr)^3+51(cr)^2+11cr+1]\, (1- cr)^{11}_+$ $\psi_3(r)$ $-62,016\, c^6\, [320 (cr)^3+197 (cr)^2+50 cr+5]\, (1- cr)^{10}_+$ $\psi_4(r)$ $3,224,832\, c^8\, [80(cr)^2+27cr+3]\, (1- cr)_+^{9}$ $\psi_5(r)$ $-354,731,520\, c^{10}\, [8cr+1]\, (1- cr)_+^{8}$ $\psi_6(r)$ $25,540,669,440\,c^{12}\,(1- cr)_+^{7}$
 $\phi_{7, 6}$ $\psi_0(r)$ $[4,096(cr)^6+7,059 (cr)^5+5,751(cr)^4+2,782(cr)^3+830(cr)^2$ $+143cr+11]\, (1- cr)^{13}_+$ $\psi_1(r)$ $-38\, c^2\, [2,048(cr)^5+2,697(cr)^4+1,644(cr)^3+566(cr)^2$ $+108cr+9]\, (1- cr)^{12}_+$ $\psi_2(r)$ $10,336\, c^4\, [128(cr)^4+121(cr)^3+51(cr)^2+11cr+1]\, (1- cr)^{11}_+$ $\psi_3(r)$ $-62,016\, c^6\, [320 (cr)^3+197 (cr)^2+50 cr+5]\, (1- cr)^{10}_+$ $\psi_4(r)$ $3,224,832\, c^8\, [80(cr)^2+27cr+3]\, (1- cr)_+^{9}$ $\psi_5(r)$ $-354,731,520\, c^{10}\, [8cr+1]\, (1- cr)_+^{8}$ $\psi_6(r)$ $25,540,669,440\,c^{12}\,(1- cr)_+^{7}$
The table shows values for $\psi_{k, i}: = \sup_{r\in[0, \infty)} |\psi_i(r)| r^k$ for the Wendland functions $\psi_0(r): = \phi_{8, 6}(cr)$ and $\psi_0(r): = \phi_{7, 6}(cr)$
 $\psi_{k, i}$ $\phi_{8, 6}$ $\phi_{7, 6}$ $\psi_{6, 6}$ $3.148511062 \cdot 10^7\cdot c^6$ $3.240130299 \cdot 10^6\cdot c^6$ $\psi_{5, 5}$ $2.363249538\cdot 10^6\cdot c^5$ $2.588617377\cdot 10^5\cdot c^5$ $\psi_{5, 4}$ $6.409097287\cdot 10^6\cdot c^6$ $6.534280933\cdot 10^5\cdot c^6$ $\psi_{4, 4}$ $1.947997580\cdot 10^5\cdot c^4$ $2.262550039\cdot 10^4\cdot c^4$ $\psi_{4, 3}$ $6.000016519\cdot 10^5 \cdot c^5$ $6.515237949\cdot 10^4 \cdot c^5$ $\psi_{4, 2}$ $2.215560450\cdot 10^6\cdot c^6$ $2.237953342\cdot 10^5\cdot c^6$ $\psi_{3, 3}$ $1.807542870\cdot 10^4\cdot c^3$ $2.219149087\cdot 10^3\cdot c^3$ $\psi_{3, 2}$ $6.618581621\cdot 10^4\cdot c^4$ $7.625999381\cdot 10^3\cdot c^4$ $\psi_{3, 1}$ $3.172360616 \cdot 10^5 \cdot c^5$ $3.414789975 \cdot 10^4 \cdot c^5$ $\psi_{3, 0}$ $3.1008\cdot 10^6\cdot c^6$ $3.1008\cdot 10^5\cdot c^6$ $\psi_{2, 2}$ $1.970990855\cdot 10^3\cdot c^2$ $2.550970282\cdot 10^2\cdot c^2$ $\psi_{2, 1}$ $9.418422390\cdot 10^3\cdot c^3$ $1.147899628\cdot 10^3\cdot c^3$ $\psi_{2, 0}$ $9.044\cdot 10^4\cdot c^4$ $1.0336\cdot 10^4\cdot c^4$ $\psi_{1, 1}$ $2.767275907\cdot 10^2\cdot c$ $3.766803387\cdot 10^1\cdot c$ $\psi_{1, 0}$ $2.66\cdot 10^3\cdot c^2$ $3.42\cdot 10^2\cdot c^2$
 $\psi_{k, i}$ $\phi_{8, 6}$ $\phi_{7, 6}$ $\psi_{6, 6}$ $3.148511062 \cdot 10^7\cdot c^6$ $3.240130299 \cdot 10^6\cdot c^6$ $\psi_{5, 5}$ $2.363249538\cdot 10^6\cdot c^5$ $2.588617377\cdot 10^5\cdot c^5$ $\psi_{5, 4}$ $6.409097287\cdot 10^6\cdot c^6$ $6.534280933\cdot 10^5\cdot c^6$ $\psi_{4, 4}$ $1.947997580\cdot 10^5\cdot c^4$ $2.262550039\cdot 10^4\cdot c^4$ $\psi_{4, 3}$ $6.000016519\cdot 10^5 \cdot c^5$ $6.515237949\cdot 10^4 \cdot c^5$ $\psi_{4, 2}$ $2.215560450\cdot 10^6\cdot c^6$ $2.237953342\cdot 10^5\cdot c^6$ $\psi_{3, 3}$ $1.807542870\cdot 10^4\cdot c^3$ $2.219149087\cdot 10^3\cdot c^3$ $\psi_{3, 2}$ $6.618581621\cdot 10^4\cdot c^4$ $7.625999381\cdot 10^3\cdot c^4$ $\psi_{3, 1}$ $3.172360616 \cdot 10^5 \cdot c^5$ $3.414789975 \cdot 10^4 \cdot c^5$ $\psi_{3, 0}$ $3.1008\cdot 10^6\cdot c^6$ $3.1008\cdot 10^5\cdot c^6$ $\psi_{2, 2}$ $1.970990855\cdot 10^3\cdot c^2$ $2.550970282\cdot 10^2\cdot c^2$ $\psi_{2, 1}$ $9.418422390\cdot 10^3\cdot c^3$ $1.147899628\cdot 10^3\cdot c^3$ $\psi_{2, 0}$ $9.044\cdot 10^4\cdot c^4$ $1.0336\cdot 10^4\cdot c^4$ $\psi_{1, 1}$ $2.767275907\cdot 10^2\cdot c$ $3.766803387\cdot 10^1\cdot c$ $\psi_{1, 0}$ $2.66\cdot 10^3\cdot c^2$ $3.42\cdot 10^2\cdot c^2$
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