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August  2019, 24(8): 4247-4269. doi: 10.3934/dcdsb.2019080

Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function

Hjörtur Björnsson 1, , Sigurdur Hafstein 1, , Peter Giesl 2, , Enrico Scalas 2, and  Skuli Gudmundsson 3,

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  August 2019 Early access  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.

Keywords: Nonlinear stochastic differential equation, Lyapunov function, Radial basis function, Basin of attraction.
Mathematics Subject Classification: Primary: 37B25, 65P40, 93E03, 93E15, 93D30; Secondary: 60H35, 65C30.
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, 2019, 24 (8) : 4247-4269. 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, . Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P. Giesl, Construction of Global Lyapunov functions using Radial Basis Functions, volume 1904 of Lecture Notes in Mathematics, Springer, Berlin, 2007.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

N. Mohammed, Grid Refinement and Verification Estimates for the RBF Construction Method of Lyapunov Functions, PhD thesis, University of Sussex, 2016. Google Scholar

[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.  Google Scholar

[12]

R. Schaback and H. Wendland, Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639.  doi: 10.1017/S0962492906270016.  Google Scholar

[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.  Google Scholar

[14]

H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.  Google Scholar

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, . Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P. Giesl, Construction of Global Lyapunov functions using Radial Basis Functions, volume 1904 of Lecture Notes in Mathematics, Springer, Berlin, 2007.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

N. Mohammed, Grid Refinement and Verification Estimates for the RBF Construction Method of Lyapunov Functions, PhD thesis, University of Sussex, 2016. Google Scholar

[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.  Google Scholar

[12]

R. Schaback and H. Wendland, Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639.  doi: 10.1017/S0962492906270016.  Google Scholar

[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.  Google Scholar

[14]

H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.  Google Scholar

Figure 1.  Above: the computed non-local Lyapunov function $ v $ for system (5.1). Below: the function $ Lv $, approximating $ -10^{-3} $
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3]">Figure 2.  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]
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Table 1.  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}_+$
$\psi_1(r)$ $-380\, c^2\, [2,431(cr)^5+2,931(cr)^4+1,638(cr)^3+518(cr)^2$
$+91cr+7]\, (1- cr)^{13}_+$
$\psi_2(r)$ $12,920\, c^4\, [1,287(cr)^4+1,108(cr)^3+426(cr)^2$
$+84cr+7]\, (1- cr)^{12}_+$
$\psi_3(r)$ $-620,160\, c^6\, [429 (cr)^3+239 (cr)^2+55 cr+5]\, (1- cr)^{11}_+$
$\psi_4(r)$ $112,869,120\, c^8\, [33(cr)^2+10cr+1]\, (1- cr)_+^{10}$
$\psi_5(r)$ $-4,966,241,280\, c^{10}\, [9cr+1]\, (1- cr)_+^{9}$
$\psi_6(r)$ $446,961,715,200\, c^{12}\, (1- cr)_+^{8}$
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$\phi_{8, 6}$
$\psi_0(r)$ $[46,189(cr)^6+73,206 (cr)^5+54,915(cr)^4+24,500(cr)^3$
$ +6,755(cr)^2+1,078cr+77]\, (1- cr)^{14}_+$
$\psi_1(r)$ $-380\, c^2\, [2,431(cr)^5+2,931(cr)^4+1,638(cr)^3+518(cr)^2$
$+91cr+7]\, (1- cr)^{13}_+$
$\psi_2(r)$ $12,920\, c^4\, [1,287(cr)^4+1,108(cr)^3+426(cr)^2$
$+84cr+7]\, (1- cr)^{12}_+$
$\psi_3(r)$ $-620,160\, c^6\, [429 (cr)^3+239 (cr)^2+55 cr+5]\, (1- cr)^{11}_+$
$\psi_4(r)$ $112,869,120\, c^8\, [33(cr)^2+10cr+1]\, (1- cr)_+^{10}$
$\psi_5(r)$ $-4,966,241,280\, c^{10}\, [9cr+1]\, (1- cr)_+^{9}$
$\psi_6(r)$ $446,961,715,200\, c^{12}\, (1- cr)_+^{8}$
Table 2.  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 $
$\phi_{7, 6}$
$\psi_0(r)$ $[4,096(cr)^6+7,059 (cr)^5+5,751(cr)^4+2,782(cr)^3+830(cr)^2$
$+143cr+11]\, (1- cr)^{13}_+$
$\psi_1(r)$ $-38\, c^2\, [2,048(cr)^5+2,697(cr)^4+1,644(cr)^3+566(cr)^2$
$+108cr+9]\, (1- cr)^{12}_+$
$\psi_2(r)$ $10,336\, c^4\, [128(cr)^4+121(cr)^3+51(cr)^2+11cr+1]\, (1- cr)^{11}_+$
$\psi_3(r)$ $-62,016\, c^6\, [320 (cr)^3+197 (cr)^2+50 cr+5]\, (1- cr)^{10}_+$
$\psi_4(r)$ $3,224,832\, c^8\, [80(cr)^2+27cr+3]\, (1- cr)_+^{9}$
$\psi_5(r)$ $-354,731,520\, c^{10}\, [8cr+1]\, (1- cr)_+^{8}$
$\psi_6(r)$ $25,540,669,440\,c^{12}\,(1- cr)_+^{7}$
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$\phi_{7, 6}$
$\psi_0(r)$ $[4,096(cr)^6+7,059 (cr)^5+5,751(cr)^4+2,782(cr)^3+830(cr)^2$
$+143cr+11]\, (1- cr)^{13}_+$
$\psi_1(r)$ $-38\, c^2\, [2,048(cr)^5+2,697(cr)^4+1,644(cr)^3+566(cr)^2$
$+108cr+9]\, (1- cr)^{12}_+$
$\psi_2(r)$ $10,336\, c^4\, [128(cr)^4+121(cr)^3+51(cr)^2+11cr+1]\, (1- cr)^{11}_+$
$\psi_3(r)$ $-62,016\, c^6\, [320 (cr)^3+197 (cr)^2+50 cr+5]\, (1- cr)^{10}_+$
$\psi_4(r)$ $3,224,832\, c^8\, [80(cr)^2+27cr+3]\, (1- cr)_+^{9}$
$\psi_5(r)$ $-354,731,520\, c^{10}\, [8cr+1]\, (1- cr)_+^{8}$
$\psi_6(r)$ $25,540,669,440\,c^{12}\,(1- cr)_+^{7}$
Table 3.  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$
$\psi_{2, 2}$ $1.970990855\cdot 10^3\cdot c^2$ $2.550970282\cdot 10^2\cdot c^2$
$\psi_{2, 1}$ $9.418422390\cdot 10^3\cdot c^3$ $1.147899628\cdot 10^3\cdot c^3$
$\psi_{2, 0}$ $9.044\cdot 10^4\cdot c^4$ $1.0336\cdot 10^4\cdot c^4$
$\psi_{1, 1}$ $2.767275907\cdot 10^2\cdot c$ $3.766803387\cdot 10^1\cdot c$
$\psi_{1, 0}$ $2.66\cdot 10^3\cdot c^2$ $3.42\cdot 10^2\cdot c^2$
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$\psi_{k, i}$ $\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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Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569
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