# American Institute of Mathematical Sciences

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

 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.

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. [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. [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. [4] H. Bucky, Stability and positive supermartingales, J. Differ. Equations, 1 (1965), 151-155. doi: 10.1016/0022-0396(65)90016-1. [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. [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. [7] P. Florchinger, Lyapunov-like techniques for stochastic stability, SIAM J. Control Optim., 33 (1995), 1151-1169. doi: 10.1137/S0363012993252309. [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. [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. [10] D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann., 32 (1888), 342-350. doi: 10.1007/BF01443605. [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. [12] R. Khasminskii, Stochastic Stability of Differential Equations, Springer, 2nd edition, 2012. [13] X. Mao, Stochastic Differential Equations and Applications, Woodhead Publishing, 2nd edition, 2008. doi: 10.1533/9780857099402. [14] J. Massera, Contributions to stability theory, Annals of Mathematics, 64 (1956), 182-206. doi: 10.2307/1969955. [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. [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. [18] B. Øksendal, Stochastic Differential Equations, An introduction with applications. Sixth edition. Universitext. Springer-Verlag, Berlin, 2003. [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. [20] B. Reznick, Uniform denominators in Hilbert's seventeenth problem, Math. Z., 220 (1995), 75-97. doi: 10.1007/BF02572604. [21] B. Reznick, Some concrete aspects of Hilbert's 17th problem, Contemporary Mathematics, 253 (2000), 251-272. [22] J. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software, 11/12 (1999), 625-653. [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. [24] U. Thygesen, A Survey of Lyapunov Techniques for Stochastic Differential Equations, IMM Technical Report, 1997. [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.

show all references

##### 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. [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. [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. [4] H. Bucky, Stability and positive supermartingales, J. Differ. Equations, 1 (1965), 151-155. doi: 10.1016/0022-0396(65)90016-1. [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. [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. [7] P. Florchinger, Lyapunov-like techniques for stochastic stability, SIAM J. Control Optim., 33 (1995), 1151-1169. doi: 10.1137/S0363012993252309. [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. [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. [10] D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann., 32 (1888), 342-350. doi: 10.1007/BF01443605. [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. [12] R. Khasminskii, Stochastic Stability of Differential Equations, Springer, 2nd edition, 2012. [13] X. Mao, Stochastic Differential Equations and Applications, Woodhead Publishing, 2nd edition, 2008. doi: 10.1533/9780857099402. [14] J. Massera, Contributions to stability theory, Annals of Mathematics, 64 (1956), 182-206. doi: 10.2307/1969955. [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. [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. [18] B. Øksendal, Stochastic Differential Equations, An introduction with applications. Sixth edition. Universitext. Springer-Verlag, Berlin, 2003. [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. [20] B. Reznick, Uniform denominators in Hilbert's seventeenth problem, Math. Z., 220 (1995), 75-97. doi: 10.1007/BF02572604. [21] B. Reznick, Some concrete aspects of Hilbert's 17th problem, Contemporary Mathematics, 253 (2000), 251-272. [22] J. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software, 11/12 (1999), 625-653. [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. [24] U. Thygesen, A Survey of Lyapunov Techniques for Stochastic Differential Equations, IMM Technical Report, 1997. [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.
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 - - - - $\text{no solution}$
 # $\omega$ $p$ $c$ $D_{11}$ $D_{22}$ $D_{33}$ $O$ 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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