Stability analysis for delay differential equations with multidelays and numerical examples

In this paper we are concerned with the asymptotic stability of the delay differential equation

where $A_0,A_k\in C^{d\times d}$ are constant complex matrices, and $x(t_{\tau_k})= (x_1(t-\tau _{k1}),x_2(t-\tau _{k2}),\dots,x_d(t-\tau _{kd}))^T,\tau _{kl}>0$ stand for $n\times d$ constant delays $(k=1,\dots,n,l=1,\dots,d)$. We obtain two criteria for stability through the evaluation of a harmonic function on the boundary of a certain region. We also get similar results for the neutral delay differential equation

where $L,M_i,$ and $N_j\in C^{d\times d}$ are constant complex matrices and $\tau _i,\tau _j^{\prime }>0$ stands for constant delays $(i=1,\dots,m$, $j=1,\dots,n)$. Numerical examples on various circumstances are shown to check our results which are more general than those already reported.