Linear pde of order one

[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.

 

[jvicens]

Hello,

Thank you for sharing your thought process and asking for help. You are correct that in order to determine whether u=-x^3_1-x_1-\sqrt{3}x_2 is a stabilizing control for the given system, you need to find a value function V that satisfies the partial differential equation (PDE) you mentioned. This PDE is known as the Hamilton-Jacobi-Bellman (HJB) equation and is a necessary condition for optimality in optimal control theory.

To solve this HJB equation, you can use various methods such as the method of characteristics, the method of characteristics with the separation of variables, or the method of characteristics with the Laplace transform. These methods are well-known in the literature and can be implemented using computer programs such as MATLAB or Mathematica.

If you are having trouble solving the system of auxiliary ODEs, you can try using the method of characteristics with the separation of variables. This method involves solving a system of two ordinary differential equations, one for each variable, and then combining the solutions to obtain the value function V. You can also try using the method of characteristics with the Laplace transform, which involves transforming the PDE into an ordinary differential equation and then solving it using standard techniques.

I hope this helps. Good luck with your research!