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Stability analysis for delay differential equations with multidelays and numerical examples

Author: Leping Sun
Journal: Math. Comp. 75 (2006), 151-165
MSC (2000): Primary 39A11
Published electronically: September 15, 2005
MathSciNet review: 2176393
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Abstract: In this paper we are concerned with the asymptotic stability of the delay differential equation

\begin{displaymath}x^{\prime }(t)=A_0x(t)+\sum_{k=1}^nA_kx(t_{\tau _k}), \end{displaymath}

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

\begin{displaymath}x^{\prime }(t)=Lx(t)+\sum_{i=1}^mM_ix(t-\tau _i)+\sum_{j=1}^nN_jx^{\prime }(t-\tau _j^{\prime }), \end{displaymath}

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.

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Additional Information

Leping Sun
Affiliation: College of Mathematical Sciences, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

Keywords: Eigenvalue, matrix norm, spectral radius, boundary criteria, asymptotic stability, harmonic function, logarithmic norm
Received by editor(s): March 2, 2003
Received by editor(s) in revised form: May 17, 2004
Published electronically: September 15, 2005
Additional Notes: The author was supported by the Shanghai Leading Academic Discipline Project, Project Number T0401.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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