# zbMATH — the first resource for mathematics

Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients. (English) Zbl 1214.34059
The 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.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K40 Neutral functional-differential equations 34K06 Linear functional-differential equations
Full Text:
##### References:
 [1] Kolmanovskii V, Myshkis A: Introduction to the Theory and Applications of Functional-Differential Equations, Mathematics and its Applications. Volume 463. Kluwer Academic Publishers, Dordrecht, the Netherlands; 1999:xvi+648. · Zbl 0917.34001 [2] Krasovskii NN: Some Problems of Theory of Stability of Motion. Fizmatgiz, Moscow, Russia; 1959. [3] Krasovskiĭ NN: Stability of Motion. Applications of Lyapunov’s Second Method to Differential systems and Equations with Delay, Translated by J. L. Brenner. Stanford University Press, Stanford, Calif, USA; 1963:vi+188. · Zbl 0109.06001 [4] Korenevskiĭ DG: Stability of Dynamical Systems Under Random Perturbations of Parameters. Algebraic Criteria. Naukova Dumka, Kiev, Ukraine; 1989:208. [5] Khusainov, DYa; Yunkova, EA, Investigation of the stability of linear systems of neutral type by the method of Lyapunov functions, Differentsial cprime nye Uravneniya, 24, 613-621, (1988) [6] Gopalsamy K: Stability and Oscillations in Delay Differential Equations of Population Pynamics, Mathematics and its Applications. Volume 74. Kluwer Academic Publishers, Dordrecht, the Netherlands; 1992:xii+501. · Zbl 0752.34039 [7] Kolmanovskiĭ V, Myshkis A: Applied Theory of Functional-Differential Equations, Mathematics and its Applications (Soviet Series). Volume 85. Kluwer Academic Publishers Group, Dordrecht, the Netherlands; 1992:xvi+234. · Zbl 0785.34005 [8] Kolmanovskiĭ V, Nosov V: Stability of Functional Differential Equations, Mathematics in Science and Engineering. Volume 180. Academic Press, Harcourt Brace Jovanovich, London, UK; 1986:xiv+ 217. [9] Mei-Gin, L, Stability analysis of neutral-type nonlinear delayed systems: an LMI approach, Journal of Zhejiang University A, 7, supplement 2, 237-244, (2006) · Zbl 1134.34328 [10] Gu K, Kharitonov VL, Chen J: Stability of Time-Delay Systems, Control Engineering. Birkhuser, Boston, Mass, USA; 2003:xx+353. · Zbl 1039.34067 [11] Liao X, Wang L, Yu P: Stability of Dynamical Systems, Monograph Series on Nonlinear Science and Complexity. Volume 5. Elsevier, Amsterdam, the Netherlands; 2007:xii+706. · Zbl 1134.34001 [12] Park, Ju-H; Won, S, A note on stability of neutral delay-differential systems, Journal of the Franklin Institute, 336, 543-548, (1999) · Zbl 0969.34066 [13] Liu, X-x; Xu, B, A further note on stability criterion of linear neutral delay-differential systems, Journal of the Franklin Institute, 343, 630-634, (2006) · Zbl 1114.34335
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.