Article Contents
Article Contents

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

• 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.

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

 Citation:

• 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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