# On stability of time delayed systems in Laplace form

Consider the following time-delayed system:

$$\dot{x}(t)=A_0 x(t)+A_1 x(t-d), \quad \quad (1)$$

where $$A_0$$ and $$A_1$$ are time-invariant matrices and $$d$$ is a constant time delay. Taking Lapace transformation leads to

$$s{X}(s)=(A_0+A_1e^{-ds})X(s).$$

Question: Suppose that we investigate the eigenvalues of $$A_0+A_1e^{-ds}$$ and derive some useful conditions based on which all eigenvalues of $$A_0+A_1e^{-ds}$$ are strictly negative. Can we conclude that system (1) is asymptotically stable under those conditions?

Example: Let $$A_0+A_1e^{-ds} = \left[\begin{matrix} Ce^{-ds} \quad D \\ M \quad -Q \end{matrix}\right]$$. Based on the schur complement it holds that if $$Ce^{-ds} + DQ^{-1}M<0$$ then $$A_0+A_1e^{-ds}$$ is negative definite. Now, we proceed the stability analysis based on conditions that satisfy $$Ce^{-ds} + DQ^{-1}M<0$$ for all $$s$$. Does this make sense?