# 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
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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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