Stability analysis for delay differential equations with multidelays and numerical examples

Source abstract

In this paper we are concerned with the asymptotic stability of the delay differential equation $$x ′ ( t ) = A 0 x ( t ) + ∑ k = 1 n A k x ( t τ k ) , x^{\prime }(t)=A_0x(t)+\sum _{k=1}^nA_kx(t_{\tau _k}),$$ where A 0 , A k ∈ C d × d A_0,A_k\in C^{d\times d} are constant complex matrices, and x ( t τ k ) = ( x 1 ( t − τ k 1 ) , x 2 ( t − τ k 2 ) , … , x d ( t − τ k d ) ) T , τ k l > 0 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 × d n\times d constant delays ( k = 1 , … , n , l = 1 , … , d ) (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 $$x ′ ( t ) = L x ( t ) + ∑ i = 1 m M i x ( t − τ i ) + ∑ j = 1 n N j x ′ ( t − τ j ′ ) , x^{\prime }(t)=Lx(t)+\sum _{i=1}^mM_ix(t-\tau _i)+\sum _{j=1}^nN_jx^{\prime }(t-\tau _j^{\prime }),$$ where L , M i , L,M_i, and N j ∈ C d × d N_j\in C^{d\times d} are constant complex matrices and τ i , τ j ′ > 0 \tau _i,\tau _j^{\prime }>0 stands for constant delays ( i = 1 , … , m (i=1,\dots ,m , j = 1 , … , n ) j=1,\dots ,n) . Numerical examples on various circumstances are shown to check our results which are more general than those already reported.