Stability analysis for delay differential equations with multidelays and numerical examples

Abstract:

In this paper we are concerned with the asymptotic stability of the delay differential equation \[ x^{\prime }(t)=A_0x(t)+\sum _{k=1}^nA_kx(t_{\tau _k}), \] 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 \[ x^{\prime }(t)=Lx(t)+\sum _{i=1}^mM_ix(t-\tau _i)+\sum _{j=1}^nN_jx^{\prime }(t-\tau _j^{\prime }), \] 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.