I really don't understand how resolve this exercise...with Lyapunoff? can someone help me? Thanks
Consider the state equation:
$$ \frac{\partial}{\partial t}x(t)= A(t)x(t), \: x(\tau)=x_0 $$
$$ A(t) = \begin{pmatrix} -1 & k(t) \\ 0 & -1 \\ \end{pmatrix}, $$
(a) Assume that $k(t)$ is constant over time, that is $k(t) = \tilde{k}$ for $t ≥ \tau$. For what value of scalar constant $\tilde{k}$ is the system exponentially stable?
(b) Is the state equation uniformly stable for all scalar functions $k(t)$? If so, provide a proof. If not, provide a counterexample.
As for the first part here is the solution :
$$X'= AX$$ Taking the Laplace on both sides , we get : $$sX(s)-x_0 = AX(s)$$ After some re-arrangement we will get it as : $$ (sI-A)X(s) = x_0$$
Here $I$ is $2\times 2$ identity matrix . Therefore putting the values of $A$, $I$ and $x_0$ . Here $x_0$ is $2\times 1$ matrix write it as \begin{bmatrix} x_{01} \\ x_{02} \\ \end{bmatrix} Now using these values of $ A,I$ and $x_0$ you will get
Taking the laplace inverse we get :
Sow we see that for any value $k$ in the real domain , it is exponentially stable .
But from my proof we can see that if $k(t)$ is $e^{t^2}$ it will not be stable