Looking for a Lyapunov function for the next system 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:
$$V_1 = \frac{1}{2}(||x||^2 + ||y||^2)$$
$$\dot{V}_1 = x^T\dot{x} + y^T\dot{y} = -x^T(A+A^T)x -y^TKy + x^T(A+K)y$$
$$V_2 = \frac{1}{2}||x-y||^2$$
$$\dot{V}_2 = (x-y)^T(\dot{x}-\dot{y})=-x^T(A+A^T)x -y^TKy + x^T(A+K)y 
-x^TKx + x^TKy + y^T(A+A^T)x - y^TAy$$
$$V_3 = \frac{1}{2}||x^Ty||^2$$
$$\dot{V}_3 = x^T\dot{y}+y^T\dot{x} = -y^T(A+A^T)y+y^TAy+x^TKx-x^TKy $$
 A: I will give a counter example to show that system can be unstable. Let
$$
X = \begin{pmatrix} x \\ y\end{pmatrix},\\
\dot X = \begin{pmatrix} -A -A^T & A \\ k I & - k I\end{pmatrix}X = \mathcal A X   
$$
If $A$ is a constant matrix then the general solution (for non-degenerate eigenvalues) is 
$$
X = e^{\mathcal A t} X_0,
$$
thus if $\mathcal A$ has a positive eigenvalue, then the system is not stable. Note that, 
$$
\det \begin{pmatrix} -A -A^T -\lambda I & A \\ k I & - k I -\lambda I\end{pmatrix} = \det( (A+ A^T +\lambda I)(k+\lambda) - Ak ),
$$
this is because the bottom two block matrices commute. This then inspires the choice (I will do 2x2 for simplicity) :
$$
A =\left(
\begin{array}{cc}
 1 & -4 \\
 4 & 1 \\
\end{array}
\right), \quad k=1,
$$
resulting in 
$$
 \det( (A+ A^T +\lambda I)(k+\lambda) - Ak ) =\lambda ^4+6 \lambda ^3+11 \lambda ^2+6 \lambda +17 =0
$$
giving the solutions 
$$\{\{\lambda \to -3.14936-1.21259 i\},\{\lambda \to -3.14936+1.21259
   i\},\{\lambda \to 0.149358\, -1.21259 i\},\{\lambda \to
   0.149358\, +1.21259 i\}\}.$$
Two of the eigenvalues have a positive real part, meaning the system is unstable.
