Stability of time-varying switched systems with time-varying delay (Q1036648)

The author first gives a generalization of Halanay's inequality, and then uses this inequality to obtain sufficient conditions for the stability of the following switched delay system \[ x'(t)=A_{\sigma_i}(t)x(t)+B_{\sigma_i}(t)x(t-\tau(t))+\int_{t_0}^t f_{\sigma_i}(t-s)x(s)ds, \] where \[ x(t)\in \mathbb R^n, A_{\sigma_i}(t), B_{\sigma_i}(t)\in \mathbb R^{n\times n}, f_{\sigma_i}\in C[\mathbb R_+,\mathbb R^{n\times n}], \sigma_i\in N_i, i=1,2,\text{ and }0\leq\tau(t)<\tau. \]