# Linear pde of order one

1. Oct 31, 2011

### matematikawan

I want to determine whether $u=-x^3_1-x_1-\sqrt{3}x_2$ is a stabilizing control for the system
$$\begin{array}{cc}\dot{x}_1=x_2\\ \dot{x}_2=x^3_1+u\end{array}$$
with cost functional
$$\frac{1}{2}\int^{\infty}_0 x^2_1 +x^2_2+u^2 \ dt.$$

After looking at some examples, I understand that I have to find a value function V that satisfies (is this correct ?? please help)
$$x_2 \frac{\partial V}{\partial x_1} +(x^3_1+u)\frac{\partial V}{\partial x_2} + \frac{1}{2}(x^2_1 +x^2_2+u^2)=0.$$

This is a linear pde of order one. The auxiliary system of ODE is
$$\frac{dx_1}{x_2}=-\frac{dx_2}{x_1+\sqrt{3}x_2}=\frac{2dV}{x^2_1 +x^2_2 +(x^3_1+x_1+\sqrt{3}x_2)^2}.$$

I think I can solve the first pair of equations because $\frac{dx_1}{x_2}=-\frac{dx_2}{x_1+\sqrt{3}x_2}$ beause it is homogeneous of degree zero.
I have trouble solving the other pair. Any suggestion or computer program is really appreciated.