Consider the following three-dimensional linear time-varying system $$\label{eq1}\tag{$\star$} \dot{x}(t)=(A \cos(\omega_1 t) + B \cos(\omega_2 t) )x(t), \ \ x(0)\in\mathbb{R}^{3}, $$ where $\omega_1$ and $\omega_2$ are positive real numbers such that $\omega_1\ne \omega_2$ and $$ A=\begin{bmatrix}a & -a & 0 \\ -a & a & 0 \\ 0 & 0 & 0\end{bmatrix}, \quad B=\begin{bmatrix}0 & 0 & 0 \\ 0 & -b & b \\ 0 & b & -b\end{bmatrix} $$ with $a,b$ positive real numbers. Notice that $A$, $B$ do not commute. However, the commutator $$ [A,B]=AB-BA = \begin{bmatrix}0 & ab & -ab \\ -ab & 0 & ab \\ ab & -ab & 0\end{bmatrix} $$ is a skew-symmetric matrix (I don't know if this can be useful though).
My questions are:
- Does there exist an explicit closed-form expression for the solution of \eqref{eq1}?
- If not, is it possible to find a bound on $\|x(t)\|$ which explicitly depends on $\omega_1$ and $\omega_2$?
I've been stuck on this problem for a while now. So I would greatly appreciate any kind of comment or help. Thanks.