# Lyapunov Function for Nonlinear System

Which Lyapunov function should I choose to show the stability (or instability) of equilibrium points? With $$k>0$$, $$K>0$$, $$\delta >0$$. The system is Hurwitz (asymptotically stable) when $$k>K$$. Standard quadratic Lyapunov function $$$$V(x_1,x_2,x_3,x_4) = \frac{k(x_1-x_2)^2}{2}+\frac{x_3^2}{2}+\frac{x_4^2}{2}-\frac{K(x_1-x_2)^2}{2}$$$$ does not work. Linearized system $$$$\begin{cases} \dot{x}_1=x_3,\\ \dot{x}_2=x_4,\\ \dot{x}_3=-k(x_1-x_2)-K x_2,\\ \dot{x}_4=k(x_1-x_2)-\delta x_4, \end{cases}$$$$ gives $$$$A = \begin{pmatrix} 0& 0& 1& 0\\ 0& 0& 0& 1\\ -k& k-K& 0& 0\\ k& -k& 0& -\delta \end{pmatrix}.$$$$ By the Lyapunov's converse theorems I know that there exists a unique symmetric positive definite matrix $$P$$ as a solution to the Lyapunov equation with $$\boldsymbol Q=\boldsymbol I$$: $$\boldsymbol{PA}+\boldsymbol{A^TP}=-\boldsymbol{Q}$$ I have tried to solve it symbolically via Matlab, but did cope with it. I have also tried to differentiate Lyapunov function without knowing explicit form for $$\boldsymbol {P}$$ but the result of $$\boldsymbol {\dot{V}} = \boldsymbol x^T \boldsymbol P \boldsymbol x$$ gives the system of 16 equations which I could not solve as well.

## 1 Answer

With the help of MATHEMATICA.

$$A = \left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -k & -k-K & 0 & 0 \\ k & -k & 0 & -\delta \\ \end{array} \right)$$

and the command

P = LyapunovSolve[A, Transpose[A], -IdentityMatrix[4]]


we obtain

$$P = \left[ \begin{array}{cccc} -\frac{\delta ^2 \left(3 k K+k (3 k+2)+K^2+K\right)+K (k+K) (2 k+K+1)}{2 \delta k (k+K) (2 k+K)} & -\frac{\delta ^2 k+\delta ^2 K+K}{4 \delta k^2+2 \delta k K} & -\frac{1}{2} & -\frac{k+1}{2 (k+K)} \\ -\frac{\delta ^2 k+\delta ^2 K+K}{4 \delta k^2+2 \delta k K} & \frac{\delta ^2 (k+K)+K (2 k+K+1)}{2 \delta (k+K) (2 k+K)} & \frac{k+1}{2 (k+K)} & -\frac{1}{2} \\ -\frac{1}{2} & \frac{k+1}{2 (k+K)} & -\frac{\frac{\delta ^2 \left(2 k^2+2 k K+k+K^2\right)}{k+K}+(k+1) K}{2 \delta k} & \frac{K}{2 \delta } \\ -\frac{k+1}{2 (k+K)} & -\frac{1}{2} & \frac{K}{2 \delta } & \frac{k K+K}{2 \delta k+2 \delta K} \\ \end{array} \right]$$