Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients.

*(English)*Zbl 1214.34059The paper deals with the linear systems of neutral differential equations with constant coefficients and a constant delay of the form

\[ \dot{x}(t)=D\dot{x}(t-\tau)+Ax(t)+Bx(t-\tau), \]

where \(t\geq 0\), \(\tau>0\), \(A,B,\) and \(D\) are \(n\times n\) constant matrices, and \(x:[-\tau,\infty)\to\mathbb{R}^n\) is a column vector-solution. The authors investigate the exponential-type stability of such systems using Lyapunov-Krasovskii type functionals. Delay-dependent conditions sufficient for the stability are formulated in terms of positivity of auxiliary matrices. Illustrative examples are shown and comparisons with known results are given.

\[ \dot{x}(t)=D\dot{x}(t-\tau)+Ax(t)+Bx(t-\tau), \]

where \(t\geq 0\), \(\tau>0\), \(A,B,\) and \(D\) are \(n\times n\) constant matrices, and \(x:[-\tau,\infty)\to\mathbb{R}^n\) is a column vector-solution. The authors investigate the exponential-type stability of such systems using Lyapunov-Krasovskii type functionals. Delay-dependent conditions sufficient for the stability are formulated in terms of positivity of auxiliary matrices. Illustrative examples are shown and comparisons with known results are given.

Reviewer: Jan Ohriska (Košice)

##### MSC:

34K20 | Stability theory of functional-differential equations |

34K40 | Neutral functional-differential equations |

34K06 | Linear functional-differential equations |

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\textit{J. Baštinec} et al., Bound. Value Probl. 2010, Article ID 956121, 20 p.. (2010; Zbl 1214.34059)

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