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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 |
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.
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. |
# | |||||||
1 | 3.0 | 0.5 | 1.6875 | 40.569 | 0.0131 | 8.6916 | |
2 | 3.0 | 1.0 | 0.6250 | 24.016 | 7.8514 | 0.0655 | |
3 | 3.0 | 1.1 | 0.2500 | 20.913 | 7.6978 | 0.1488 | |
4 | 3.0 | 1.2 | - | - | - | - | no solution |
5 | 4.0 | 0.1 | 1.0000 | 0.0296 | 8.463 | 45.200 | |
6 | 3.5 | 0.1 | 0.6600 | 45.967 | 0.0093 | 7.8397 | |
7 | 3.0 | 0.1 | 0.25 | 47.020 | 0.0193 | 7.7424 | |
8 | 2.75 | 0.1 | 0.05 | 47.486 | 0.0159 | 7.5072 | |
9 | 2.5 | 0.1 | - | - | - | - |
# | |||||||
1 | 3.0 | 0.5 | 1.6875 | 40.569 | 0.0131 | 8.6916 | |
2 | 3.0 | 1.0 | 0.6250 | 24.016 | 7.8514 | 0.0655 | |
3 | 3.0 | 1.1 | 0.2500 | 20.913 | 7.6978 | 0.1488 | |
4 | 3.0 | 1.2 | - | - | - | - | no solution |
5 | 4.0 | 0.1 | 1.0000 | 0.0296 | 8.463 | 45.200 | |
6 | 3.5 | 0.1 | 0.6600 | 45.967 | 0.0093 | 7.8397 | |
7 | 3.0 | 0.1 | 0.25 | 47.020 | 0.0193 | 7.7424 | |
8 | 2.75 | 0.1 | 0.05 | 47.486 | 0.0159 | 7.5072 | |
9 | 2.5 | 0.1 | - | - | - | - |
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