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) = ktilde for t ≥ tau. For what value of scalar constant ktilde 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.
For the first part, recall some basic facts about systems of linear, constant-coefficient differential equations:
If $k \equiv \tilde k$ is constant, one solution will be $\xi e^{\lambda t}$ where $\lambda$ is (the only) eigenvalue of $A$, and $\xi$ is the corresponding eigenvector.
For most values of $\tilde k$, the other solution will be given by $\left( t\xi + \eta\right) e^{\lambda t}$ where $\eta$ solves
$$ (A -\lambda I)\eta = \xi $$
otherwise (for one particular value of $\tilde k$), the other solution is $\eta e^{\lambda t}$, where $\eta$ is another eigenvector of $A$ that is linearly independent w.r.t. $\xi$.
So, given this, what happens to solutions as $t \to \infty$?
For the second part, you'll just have to play around. Try using this: http://cs.jsu.edu/~leathrum/Mathlets/diffeq2.html#instr with various guesses for $k(t)$ and see what happens.