I am really stuck looking for a Lypaunov candidate for the next system (which in simulation is stable).
$$ \dot{x} = -(A+A^T)x + Ay \\ \dot{y} = K(x-y) $$
where x and y are vectors in R^3, A is a time varying matrix such that $A+A^T > 0$, so $x^TAx > 0$. And $K$ is $kI$, where $I$ is the identity matrix and $k$ a positive real constant.
I have tried as Lyapunov candidates $||x||^2+||y||^2$, $||x-y||^2$, $||x+y||^2$ and $||x^Ty||^2$, but I always find cross terms in the derivative that I can not eliminate. Any other clues or hints?
Many thanks in advance
some computations in order to follow the problem:
$$V_1 = \frac{1}{2}(||x||^2 + ||y||^2)$$ $$\dot{V}_1 = x^T\dot{x} + y^T\dot{y} = -x^T(A+A^T)x -y^TKy + x^T(A+K)y$$ $$V_2 = \frac{1}{2}||x-y||^2$$ $$\dot{V}_2 = (x-y)^T(\dot{x}-\dot{y})=-x^T(A+A^T)x -y^TKy + x^T(A+K)y -x^TKx + x^TKy + y^T(A+A^T)x - y^TAy$$ $$V_3 = \frac{1}{2}||x^Ty||^2$$ $$\dot{V}_3 = x^T\dot{y}+y^T\dot{x} = -y^T(A+A^T)y+y^TAy+x^TKx-x^TKy $$