American Institute of Mathematical Sciences

Advanced
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 and 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, .

[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.

Figure 1.  Above: the computed non-local Lyapunov function $ v $ for system (5.1). Below: the function $ Lv $, approximating $ -10^{-3} $
Figure Options

Download full-size image

Download as PowerPoint slide

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]
Figure Options

Download full-size image

Download as PowerPoint slide

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}_+$
$\psi_1(r)$ $-380\, c^2\, [2,431(cr)^5+2,931(cr)^4+1,638(cr)^3+518(cr)^2$
$+91cr+7]\, (1- cr)^{13}_+$
$\psi_2(r)$ $12,920\, c^4\, [1,287(cr)^4+1,108(cr)^3+426(cr)^2$
$+84cr+7]\, (1- cr)^{12}_+$
$\psi_3(r)$ $-620,160\, c^6\, [429 (cr)^3+239 (cr)^2+55 cr+5]\, (1- cr)^{11}_+$
$\psi_4(r)$ $112,869,120\, c^8\, [33(cr)^2+10cr+1]\, (1- cr)_+^{10}$
$\psi_5(r)$ $-4,966,241,280\, c^{10}\, [9cr+1]\, (1- cr)_+^{9}$
$\psi_6(r)$ $446,961,715,200\, c^{12}\, (1- cr)_+^{8}$
Table Options

Download as excel

$\phi_{8, 6}$
$\psi_0(r)$ $[46,189(cr)^6+73,206 (cr)^5+54,915(cr)^4+24,500(cr)^3$
$ +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 $
$\phi_{7, 6}$
$\psi_0(r)$ $[4,096(cr)^6+7,059 (cr)^5+5,751(cr)^4+2,782(cr)^3+830(cr)^2$
$+143cr+11]\, (1- cr)^{13}_+$
$\psi_1(r)$ $-38\, c^2\, [2,048(cr)^5+2,697(cr)^4+1,644(cr)^3+566(cr)^2$
$+108cr+9]\, (1- cr)^{12}_+$
$\psi_2(r)$ $10,336\, c^4\, [128(cr)^4+121(cr)^3+51(cr)^2+11cr+1]\, (1- cr)^{11}_+$
$\psi_3(r)$ $-62,016\, c^6\, [320 (cr)^3+197 (cr)^2+50 cr+5]\, (1- cr)^{10}_+$
$\psi_4(r)$ $3,224,832\, c^8\, [80(cr)^2+27cr+3]\, (1- cr)_+^{9}$
$\psi_5(r)$ $-354,731,520\, c^{10}\, [8cr+1]\, (1- cr)_+^{8}$
$\psi_6(r)$ $25,540,669,440\,c^{12}\,(1- cr)_+^{7}$
Table Options

Download as excel

$\phi_{7, 6}$
$\psi_0(r)$ $[4,096(cr)^6+7,059 (cr)^5+5,751(cr)^4+2,782(cr)^3+830(cr)^2$
$+143cr+11]\, (1- cr)^{13}_+$
$\psi_1(r)$ $-38\, c^2\, [2,048(cr)^5+2,697(cr)^4+1,644(cr)^3+566(cr)^2$
$+108cr+9]\, (1- cr)^{12}_+$
$\psi_2(r)$ $10,336\, c^4\, [128(cr)^4+121(cr)^3+51(cr)^2+11cr+1]\, (1- cr)^{11}_+$
$\psi_3(r)$ $-62,016\, c^6\, [320 (cr)^3+197 (cr)^2+50 cr+5]\, (1- cr)^{10}_+$
$\psi_4(r)$ $3,224,832\, c^8\, [80(cr)^2+27cr+3]\, (1- cr)_+^{9}$
$\psi_5(r)$ $-354,731,520\, c^{10}\, [8cr+1]\, (1- cr)_+^{8}$
$\psi_6(r)$ $25,540,669,440\,c^{12}\,(1- cr)_+^{7}$
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$
$\psi_{2, 2}$ $1.970990855\cdot 10^3\cdot c^2$ $2.550970282\cdot 10^2\cdot c^2$
$\psi_{2, 1}$ $9.418422390\cdot 10^3\cdot c^3$ $1.147899628\cdot 10^3\cdot c^3$
$\psi_{2, 0}$ $9.044\cdot 10^4\cdot c^4$ $1.0336\cdot 10^4\cdot c^4$
$\psi_{1, 1}$ $2.767275907\cdot 10^2\cdot c$ $3.766803387\cdot 10^1\cdot c$
$\psi_{1, 0}$ $2.66\cdot 10^3\cdot c^2$ $3.42\cdot 10^2\cdot c^2$
Table Options

Download as excel

$\psi_{k, i}$ $\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$
$\psi_{2, 2}$ $1.970990855\cdot 10^3\cdot c^2$ $2.550970282\cdot 10^2\cdot c^2$
$\psi_{2, 1}$ $9.418422390\cdot 10^3\cdot c^3$ $1.147899628\cdot 10^3\cdot c^3$
$\psi_{2, 0}$ $9.044\cdot 10^4\cdot c^4$ $1.0336\cdot 10^4\cdot c^4$
$\psi_{1, 1}$ $2.767275907\cdot 10^2\cdot c$ $3.766803387\cdot 10^1\cdot c$
$\psi_{1, 0}$ $2.66\cdot 10^3\cdot c^2$ $3.42\cdot 10^2\cdot c^2$
[1]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101
[2]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure and Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569
[3]

Sigurdur Hafstein, Skuli Gudmundsson, Peter Giesl, Enrico Scalas. Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 939-956. doi: 10.3934/dcdsb.2018049
[4]

Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control and Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015
[5]

Robert Baier, Lars Grüne, Sigurđur Freyr Hafstein. Linear programming based Lyapunov function computation for differential inclusions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 33-56. doi: 10.3934/dcdsb.2012.17.33
[6]

Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57
[7]

Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453
[8]

Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2399-2416. doi: 10.3934/dcdss.2020403
[9]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[10]

Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control and Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501
[11]

Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139
[12]

Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3575-3589. doi: 10.3934/dcdsb.2017180
[13]

Peter Giesl. Construction of a finite-time Lyapunov function by meshless collocation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2387-2412. doi: 10.3934/dcdsb.2012.17.2387
[14]

Fabio Camilli, Lars Grüne. Characterizing attraction probabilities via the stochastic Zubov equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 457-468. doi: 10.3934/dcdsb.2003.3.457
[15]

Piermarco Cannarsa, Peter R. Wolenski. Semiconcavity of the value function for a class of differential inclusions. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 453-466. doi: 10.3934/dcds.2011.29.453
[16]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076
[17]

Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111
[18]

Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial and Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353
[19]

Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial and Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345
[20]

Yves Edel, Alexander Pott. A new almost perfect nonlinear function which is not quadratic. Advances in Mathematics of Communications, 2009, 3 (1) : 59-81. doi: 10.3934/amc.2009.3.59

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (267)
  • HTML views (265)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy