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

Sigurdur Hafstein; Skuli Gudmundsson; Peter Giesl; Enrico Scalas

doi:10.3934/dcdsb.2018049

Article Contents

2018, Volume 23, Issue 2: 939-956. Doi: 10.3934/dcdsb.2018049

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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

Received: December 31, 2016

Revised: July 31, 2017

Early access: November 2017

Published: June 2018

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Abstract

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.

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

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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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