Consider a linear time-varying system: \begin{equation} \dot x(t) = A(t) x(t), \tag{$*$} \end{equation} where $A(t)$ is a time-varying block matrix defined as $$ A(t) = \begin{bmatrix} 0 & I\\ -\gamma M(t) & -\gamma I \end{bmatrix}, $$ and $0 \prec \alpha I \preceq M(t) \preceq \beta I$, $M(t)$ is continuously differentiable for all $t\geq0$.
My question is: Can we say anything about convergence rate of $x$ towards $0$?
I've been considering differential Lyapunov inequalities in order to prove the convergence rate. According to page 117 in [1], the linear state equation $(*)$ is uniformaly exponentially stable if there exists an $n\times n$ matrix function $P(t)$ that for all $t$ is symmetric, continuously differentiable, and such that $$\eta I \preceq P(t) \preceq \rho I,$$ $$A^\top(t)P(t) + P(t)A(t) + \dot P(t) \preceq -v I.$$
In a recent paper, [2] shows that $$\dot P(t) + A^\top(t)P(t) + P(t)A(t) \preceq 2\mu(t)P(t),$$ where $\mu(t)$ is uniformly exponentially stable can be used to show exponential convergence rate. The same paper also shows that if $A(t)$ is an blocked upper-triangular matrix, then it's much easier to prove exponential stability (which is not the case in this example, since $A(t)$ here is not a blocked upper-triangular matrix).
Any ideas on how to find such a matrix $P(t)$, or other suggestions?
[1] Rugh, Wilson J., Linear system theory., Upper Saddle River, NJ: Prentice Hall. xv, 581 p. (1996). ZBL0892.93002.
[2] Zhou, Bin, On asymptotic stability of linear time-varying systems, Automatica 68, 266-276 (2016). ZBL1334.93152.
