Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming

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March 2018, 23(2): 939-956. doi: 10.3934/dcdsb.2018049

Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming

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

1.	Faculty of Physical Sciences, University of Iceland, Dunhagi 5, IS-107 Reykjavik, Iceland
2.	Svensk Exportkredit, Klarabergsviadukten 61-63, 111 64 Stockholm, Sweden
3.	Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom

Received January 2017 Revised August 2017 Published December 2017

We study the global asymptotic stability in probability of the zero solution of linear stochastic differential equations with constant coefficients. We develop a sum-of-squares program that verifies whether a parameterized candidate Lyapunov function is in fact a global Lyapunov function for such a system. Our class of candidate Lyapunov functions are naturally adapted to the problem. We consider functions of the form $V(\mathbf{x}) = \|\mathbf{x}\|_Q^p: = (\mathbf{x}^\top Q\mathbf{x})^{\frac{p}{2}}$, where the parameters are the positive definite matrix $Q$ and the number $p>0$. We give several examples of our proposed method and show how it improves previous results.

Keywords: Stochastic differential equation, Lyapunov function, stability, basin of attraction, dynamical system, numerical method.

Mathematics Subject Classification: Primary: 60H10, 60H35; Secondary: 37C75, 12D15.

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

References:

[1]	J. Anderson and A. Papachristodoulou, Advances in computational Lyapunov analysis using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2361-2381. doi: 10.3934/dcdsb.2015.20.2361. Google Scholar
[2]	S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, volume 15 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. Google Scholar
[3]	C. Briat, Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints, Automatica, 74 (2016), 279-287. doi: 10.1016/j.automatica.2016.08.001. Google Scholar
[4]	H. Bucky, Stability and positive supermartingales, J. Differ. Equations, 1 (1965), 151-155. doi: 10.1016/0022-0396(65)90016-1. Google Scholar
[5]	J. Fisher and R. Bhattacharya, Stability analysis of stochastic systems using polynomial chaos, Proceedings of the American Control Conference 11-13 June 2008, (2008), 4250-4255. doi: 10.1109/ACC.2008.4587161. Google Scholar
[6]	J. Fisher and R. Bhattacharya, Linear quadratic regulation of systems with stochastic parameter uncertainties, Automatica J. IFAC, 45 (2009), 2831-2841. doi: 10.1016/j.automatica.2009.10.001. Google Scholar
[7]	P. Florchinger, Lyapunov-like techniques for stochastic stability, SIAM J. Control Optim., 33 (1995), 1151-1169. doi: 10.1137/S0363012993252309. Google Scholar
[8]	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
[9]	L. Grüne and F. Camilli, 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
[10]	D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann., 32 (1888), 342-350. doi: 10.1007/BF01443605. Google Scholar
[11]	R. Kamyar and M. Peet, Polynomial optimization with applications to stability analysis and control -an alternative to sum of squares, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2383-2417. doi: 10.3934/dcdsb.2015.20.2383. Google Scholar
[12]	R. Khasminskii, Stochastic Stability of Differential Equations, Springer, 2nd edition, 2012. Google Scholar
[13]	X. Mao, Stochastic Differential Equations and Applications, Woodhead Publishing, 2nd edition, 2008. doi: 10.1533/9780857099402. Google Scholar
[14]	J. Massera, Contributions to stability theory, Annals of Mathematics, 64 (1956), 182-206. doi: 10.2307/1969955. Google Scholar
[15]	T. Mikosch, G. Samorodnitsky and L. Tafakori, Fractional moments of solutions to stochastic recurrence equations, Journal of Applied Probability, 50 (2013), 969-982. doi: 10.1017/S0021900200013747. Google Scholar
[16]	T. S. Motzkin, The arithmetic-geometric inequality, In Inequalities (Proc. Sympos. WrightPatterson Air Force Base, Ohio, 1965), Academic Press, New York, 1967.
[17]	R. Nigmatullin, The statistics of the fractional moments: Is there any chance to "read quantitatively" any randomness?, Signal Processing, 86 (2006), 2529-2547. doi: 10.1016/j.sigpro.2006.02.003. Google Scholar
[18]	B. Øksendal, Stochastic Differential Equations, An introduction with applications. Sixth edition. Universitext. Springer-Verlag, Berlin, 2003. Google Scholar
[19]	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
[20]	B. Reznick, Uniform denominators in Hilbert's seventeenth problem, Math. Z., 220 (1995), 75-97. doi: 10.1007/BF02572604. Google Scholar
[21]	B. Reznick, Some concrete aspects of Hilbert's 17th problem, Contemporary Mathematics, 253 (2000), 251-272. Google Scholar
[22]	J. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software, 11/12 (1999), 625-653. Google Scholar
[23]	T. Tamba and M. Lemmon, Stochastic reachability of jump-diffusion process using sum of squares optimization, unpublished, see https://www3.nd.edu/~lemmon/projects/NSF-12-520/pubs/2014/TL_TAC14_2col.pdf, 2014.Google Scholar
[24]	U. Thygesen, A Survey of Lyapunov Techniques for Stochastic Differential Equations, IMM Technical Report, 1997.Google Scholar
[25]	VanAntwerp and Braatz, A tutorial on linear and bilinear matrix inequalities, Journal of Process Control, 10 (2000), 363-385. doi: 10.1016/S0959-1524(99)00056-6. Google Scholar

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References:

[1]	J. Anderson and A. Papachristodoulou, Advances in computational Lyapunov analysis using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2361-2381. doi: 10.3934/dcdsb.2015.20.2361. Google Scholar
[2]	S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, volume 15 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. Google Scholar
[3]	C. Briat, Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints, Automatica, 74 (2016), 279-287. doi: 10.1016/j.automatica.2016.08.001. Google Scholar
[4]	H. Bucky, Stability and positive supermartingales, J. Differ. Equations, 1 (1965), 151-155. doi: 10.1016/0022-0396(65)90016-1. Google Scholar
[5]	J. Fisher and R. Bhattacharya, Stability analysis of stochastic systems using polynomial chaos, Proceedings of the American Control Conference 11-13 June 2008, (2008), 4250-4255. doi: 10.1109/ACC.2008.4587161. Google Scholar
[6]	J. Fisher and R. Bhattacharya, Linear quadratic regulation of systems with stochastic parameter uncertainties, Automatica J. IFAC, 45 (2009), 2831-2841. doi: 10.1016/j.automatica.2009.10.001. Google Scholar
[7]	P. Florchinger, Lyapunov-like techniques for stochastic stability, SIAM J. Control Optim., 33 (1995), 1151-1169. doi: 10.1137/S0363012993252309. Google Scholar
[8]	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
[9]	L. Grüne and F. Camilli, 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
[10]	D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann., 32 (1888), 342-350. doi: 10.1007/BF01443605. Google Scholar
[11]	R. Kamyar and M. Peet, Polynomial optimization with applications to stability analysis and control -an alternative to sum of squares, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2383-2417. doi: 10.3934/dcdsb.2015.20.2383. Google Scholar
[12]	R. Khasminskii, Stochastic Stability of Differential Equations, Springer, 2nd edition, 2012. Google Scholar
[13]	X. Mao, Stochastic Differential Equations and Applications, Woodhead Publishing, 2nd edition, 2008. doi: 10.1533/9780857099402. Google Scholar
[14]	J. Massera, Contributions to stability theory, Annals of Mathematics, 64 (1956), 182-206. doi: 10.2307/1969955. Google Scholar
[15]	T. Mikosch, G. Samorodnitsky and L. Tafakori, Fractional moments of solutions to stochastic recurrence equations, Journal of Applied Probability, 50 (2013), 969-982. doi: 10.1017/S0021900200013747. Google Scholar
[16]	T. S. Motzkin, The arithmetic-geometric inequality, In Inequalities (Proc. Sympos. WrightPatterson Air Force Base, Ohio, 1965), Academic Press, New York, 1967.
[17]	R. Nigmatullin, The statistics of the fractional moments: Is there any chance to "read quantitatively" any randomness?, Signal Processing, 86 (2006), 2529-2547. doi: 10.1016/j.sigpro.2006.02.003. Google Scholar
[18]	B. Øksendal, Stochastic Differential Equations, An introduction with applications. Sixth edition. Universitext. Springer-Verlag, Berlin, 2003. Google Scholar
[19]	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
[20]	B. Reznick, Uniform denominators in Hilbert's seventeenth problem, Math. Z., 220 (1995), 75-97. doi: 10.1007/BF02572604. Google Scholar
[21]	B. Reznick, Some concrete aspects of Hilbert's 17th problem, Contemporary Mathematics, 253 (2000), 251-272. Google Scholar
[22]	J. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software, 11/12 (1999), 625-653. Google Scholar
[23]	T. Tamba and M. Lemmon, Stochastic reachability of jump-diffusion process using sum of squares optimization, unpublished, see https://www3.nd.edu/~lemmon/projects/NSF-12-520/pubs/2014/TL_TAC14_2col.pdf, 2014.Google Scholar
[24]	U. Thygesen, A Survey of Lyapunov Techniques for Stochastic Differential Equations, IMM Technical Report, 1997.Google Scholar
[25]	VanAntwerp and Braatz, A tutorial on linear and bilinear matrix inequalities, Journal of Process Control, 10 (2000), 363-385. doi: 10.1016/S0959-1524(99)00056-6. Google Scholar

Table 1. Results of checking whether $P_c(\mathbf{x})$ for system (14) can be written as SOS. No solution means that even for $c = 0$ SOSTOOLS was not able to write $P_c(\mathbf{x})$ as SOS. In all the experiments we set $\sigma = 2.0$. In experiments $\#1$ to $\#4$ we set $k = 1.5$ and in experiments $\#5$ to $\#9$ we set $k = 0.9$.

#	$\omega$	$p$	$c$	$D_{11}$	$D_{22}$	$D_{33}$	$O$
1	3.0	0.5	1.6875	40.569	0.0131	8.6916	$\begin{pmatrix} -0.2380& 0.1543& 0.9590 \\ -0.8442&0.4555&-0.2827 \\ 0.4804&0.8768& -0.0218\\ \end{pmatrix}$
2	3.0	1.0	0.6250	24.016	7.8514	0.0655	$\begin{pmatrix} -0.3602& 0.1540& 0.9200 \\ -0.7665& 0.5134& -0.3860 \\ 0.5318&0.8442&0.0669\\ \end{pmatrix}$
3	3.0	1.1	0.2500	20.913	7.6978	0.1488	$\begin{pmatrix}-0.4043& 0.1477& 0.9026\\ -0.7377& 0.5308&-0.4173\\ 0.5407&0.8346& 0.1057 \\ \end{pmatrix}$
4	3.0	1.2	-	-	-	-	no solution
5	4.0	0.1	1.0000	0.0296	8.463	45.200	$\begin{pmatrix} -0.8104&0.4716&-0.3476\\ 0.4819&0.8740&0.0621\\ -0.3331&0.1172&0.9356\\ \end{pmatrix}$
6	3.5	0.1	0.6600	45.967	0.0093	7.8397	$\begin{pmatrix} -0.3094&0.1190& 0.9435\\ -0.8109&0.4852&-0.3271\\ 0.4967&0.8663& 0.0536\\ \end{pmatrix}$
7	3.0	0.1	0.25	47.020	0.0193	7.7424	$\begin{pmatrix} -0.2913&0.1212& 0.9489\\ -0.8304&0.4605& -0.3137\\ 0.4750& 0.8793& 0.0335\\ \end{pmatrix}$
8	2.75	0.1	0.05	47.486	0.0159	7.5072	$\begin{pmatrix} -0.2806& 0.1218& 0.9521\\ -0.8335& 0.4609&-0.3046\\ 0.4759& 0.8791& 0.0278\\ \end{pmatrix}$
9	2.5	0.1	-	-	-	-	$\text{no solution}$

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#	$\omega$	$p$	$c$	$D_{11}$	$D_{22}$	$D_{33}$	$O$
1	3.0	0.5	1.6875	40.569	0.0131	8.6916	$\begin{pmatrix} -0.2380& 0.1543& 0.9590 \\ -0.8442&0.4555&-0.2827 \\ 0.4804&0.8768& -0.0218\\ \end{pmatrix}$
2	3.0	1.0	0.6250	24.016	7.8514	0.0655	$\begin{pmatrix} -0.3602& 0.1540& 0.9200 \\ -0.7665& 0.5134& -0.3860 \\ 0.5318&0.8442&0.0669\\ \end{pmatrix}$
3	3.0	1.1	0.2500	20.913	7.6978	0.1488	$\begin{pmatrix}-0.4043& 0.1477& 0.9026\\ -0.7377& 0.5308&-0.4173\\ 0.5407&0.8346& 0.1057 \\ \end{pmatrix}$
4	3.0	1.2	-	-	-	-	no solution
5	4.0	0.1	1.0000	0.0296	8.463	45.200	$\begin{pmatrix} -0.8104&0.4716&-0.3476\\ 0.4819&0.8740&0.0621\\ -0.3331&0.1172&0.9356\\ \end{pmatrix}$
6	3.5	0.1	0.6600	45.967	0.0093	7.8397	$\begin{pmatrix} -0.3094&0.1190& 0.9435\\ -0.8109&0.4852&-0.3271\\ 0.4967&0.8663& 0.0536\\ \end{pmatrix}$
7	3.0	0.1	0.25	47.020	0.0193	7.7424	$\begin{pmatrix} -0.2913&0.1212& 0.9489\\ -0.8304&0.4605& -0.3137\\ 0.4750& 0.8793& 0.0335\\ \end{pmatrix}$
8	2.75	0.1	0.05	47.486	0.0159	7.5072	$\begin{pmatrix} -0.2806& 0.1218& 0.9521\\ -0.8335& 0.4609&-0.3046\\ 0.4759& 0.8791& 0.0278\\ \end{pmatrix}$
9	2.5	0.1	-	-	-	-	$\text{no solution}$

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