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I have a dynamical system which has the following form:
$\dot x=\mathcal F_1(m_1)x+\mathcal F_2(m_2)x$.
My objective is to find the parameters $m_1$ and $m_2$ via LMI (linear matrix inequality) using the Lyapunov function $V=x^TPx$ where $x$ is the state and $P$ is a positive definite matrix. The problem that I can't find a feasible solution with a single matrix $P$. I tried to solve the problem with tow steps:

  1. I consider that $\dot x=\mathcal F_1(m_1)x+\omega_1$ then using the Lyapunov function $V_1=x^TP_1x$ (of course using the mathematical background) I can find the first parameter $m_1$;
  2. I consider that $\dot x=\mathcal F_2(m_2)x+\omega_2$ and as in the first step I can find the second parameter $m_2$ using the Lyapunov function $V_2=x^TP_2x$ and using them in the simulation they work very well.

My questions are about:

  1. Is what I did logic?, is it reasonable?
  2. $m_1$ and $m_2$ were found independently how can I guarantee the stability of the system using $m_1$ and $m_2$ in other words how can I prove the stability of the system?
  3. It is well known that the Lyapunov function is also so called the energy of the system so can I say that $V=\frac{1}{2}(V_1+V_2)$ is also the energy of the system?
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