On the Asymptotic Stability of the Solutions of Functional Differential Equations with Infinite Delay

On the asymptotic stability in terms of two measures for functional differential equations

B.S. Razumikhin's concept in the qualitative theory of systems delay is 
clarified and discussed. Various ways of improvements of stability conditions are 
considered. The author shows that the guiding role of Lyapunov functions and 
demonstrates Razumikhin's method as a practical case of continuous version of 
the mathematical induction. Several examples demonstrate the obtained results.

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Razumikhin's method in the qualitative theory of processes with delay

On the convergence of solutions of functional differential equations with infinite delay

Sufficient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous FDE's whose right-hand sides can be unbounded functions of the time. The theorems are based upon Lyapunov-Krasovskii functionals whose derivatives with respect to the equations are negative semidefinite and can vanish at long intervals. The functionals and their derivatives are estimated by either x(t), the norm of the instantaneous value of the solutions or ∥x t ∥ 2 , the L 2 -norm of the phase segment of the solutions. Examples are given to show that the conditions are sharp, and the main theorems with the two different types of estimates are independent and improve earlier results. The theorems are applied to linear and nonlinear retarded FDE's with one delay and with distributed delays.

On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals

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J. Terjéki

Papers

On the asymptotic stability of solutions of functional-differential equations