# Stability test for LTV systems by differential Lyapunov inequalities

Consider a linear time-varying system: $$$$\dot x(t) = A(t) x(t), \tag{*}$$$$ where $$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.