Stability of time-delay systems via the Razumikhin method

The authors consider the time delay systems both with and without a perturbation term $$\begin{aligned} \dot{x}(t)=-Dx(t)+ C\int _{t-h}^{t}x(s)\mathrm{d}s + P(t,x(t)) \end{aligned}$$ and $$\begin{aligned} \dot{x}(t)= Dx(t) + C\int _{t-h}^{t}x(s)\mathrm{d}s, \end{aligned}$$ where $$x(t)\in {\mathbb {R}}^n$$ is the state vector, D and $$C\in {\mathbb {R}}^{n\times n}$$ are constant matrices, $$P\in C({\mathbb {R}}^{+}\times {\mathbb {R}}^{n},{\mathbb {R}}^{n})$$ , and $$h>0$$ is a constant time delay. They use the Razumikhin method to obtain some new conditions for the uniform asymptotic stability, instability, and exponential stability of the zero solution, the square integrability of the norms of all solutions of the unperturbed equation, and the boundedness of solutions of the perturbed equation. In the process, they are able to give a much simpler version of a recent result by Tian et al. (Appl Math Lett 101:106058, 2020).