Matrix-valued functionals approach for stability analysis of functional differential equations. (Q1426709)

The author investigates the stability and the uniform asymptotic stability for some nonautonomous functional-differential equations with finite delay. The approach proposed in this work is based on the Lyapunov matrix-valued functional and the so-called wedge functions. The main result of this work (is Theorem 4.1 which) gives sufficient conditions in order to have stability, uniform stability and uniform asymptotic stability of the equilibrium state at zero. The results extend previous ones due to \textit{T. A. Burton} [Stability and periodic solutions of ordinary and functional-differential equations. Mathematics in Science and Engineering; Vol. 178. Orlando ect.: Academic Press (1985; Zbl 0635.34001)]; \textit{R. H. Hering} [J. Math. Anal. Appl. 180, No. 1, 160--173 (1993; Zbl 0804.34068)]; \textit{N. N. Krasovskii} [Stability of Motion, Stanford, Calif.: Stanford University Press (1963; Zbl 0109.06001)] and \textit{Bo Zhang} [Differ. Integral Equ. 9, No. 1, 199--208 (1996; Zbl 0840.34086)].