Stability of linear neutral delay-differential systems. (English) Zbl 0669.34074

Li, Li-Ming

Bull. Aust. Math. Soc. 38, No. 3, 339-344 (1988).

Suppose that A, B and C are real \(n\times n\) matrices and \(\mu (A)=\lim_{h\to 0+}h^{-1}[\| I-hA\| -1].\) The paper deals with the system \((i)\quad \dot x(t)=Ax(t)+Bx(t-\tau)+C\dot x(t-\tau),\) where \(\tau >0\). Suppose that \(\| C\| <1\) and \(\mu (A)+(\| B\| +\| A\| \| C\|)(1-\| C\|)^{-1}<0.\) Then there exist \(M\geq 1\), \(\alpha >0\) such that \(\| x(t,\Phi)\| +\| \dot x(t,\Phi)\| \leq 2M\cdot \| \Phi \| \cdot e^{-\alpha t},\) \(t\in [0,+\infty)\), where x(\(\cdot,\Phi)\) is a solution of (i) with \(x(t)=\Phi(t),\) \(\dot x(t)={\dot \Phi}(t)\) on [-\(\tau\),0]. A method of differential inequalities is used.

Reviewer: Z.Kamont

MSC:

34K20	Stability theory of functional-differential equations
34A40	Differential inequalities involving functions of a single real variable
34D05	Asymptotic properties of solutions to ordinary differential equations

Keywords:

method of differential inequalities

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