Lyapunov's direct method for linear systems of functional-differential equations in Sobolev space

R. K. Romanovsky, E. M. Nazaruk

Omsk State Technical University, Omsk, Russia

Abstract: We establish a criterion for exponential stability in the $\mathrm H^1$-topology in terms of operator inequalities for a linear FDE system of retarded type by Lyapunov's direct method. As a corollary, some sufficient condition of exponential stability in terms of the matrix specifying the Stieltjes integral is obtained in the autonomous case. A few examples illustrating the results are exhibited.

Keywords: reduction to a difference equation in a Sobolev space, matrix realization of operators in $\mathrm H^1(0,1)$, stability in $\mathrm H^1$-topology, Lyapunov functional.

UDC: 517.9

Received: 17.11.2012
Revised: 30.03.2014

 English version: Siberian Mathematical Journal, 2014, 55:4, 696–705

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