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

Abstract

Abstract. 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-Krasovskiȋ 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 xt 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.

Stability for nonautonomous linear differential systems with infinite delay

“…For the last few years, there has been a renewed interest in the analysis of stability of nonautonomous linear DDEs, and several methods and tools have been proposed, see e.g. [2,4,5,14,15,17,18,21,28,29] and references therein. The main goal of this paper is to obtain new explicit sufficient conditions for the asymptotic and exponential asymptotic stability of a general linear system (1.1), which improve and generalize some criteria in recent literature.…”

“…One of the goals of the present paper is to remove this constraint. Recently, Hatvani [21] and Györi and Horváth [17] achieved sharper results for the asymptotic stability of scalar differential equations x ′ (t) = −d(t)x(t) + β(t)x(t − τ (t)) and inequalities x ′ (t) ≤ −d(t)x(t) + β(t)x(t − τ (t)), respectively, without the a priori requirement of having bounded coefficients. We shall show that some of the results in [17,18,21,28] are a simple consequence of the stability criteria established here for ndimensional linear DDEs.…”

“…Recently, Hatvani [21] and Györi and Horváth [17] achieved sharper results for the asymptotic stability of scalar differential equations x ′ (t) = −d(t)x(t) + β(t)x(t − τ (t)) and inequalities x ′ (t) ≤ −d(t)x(t) + β(t)x(t − τ (t)), respectively, without the a priori requirement of having bounded coefficients. We shall show that some of the results in [17,18,21,28] are a simple consequence of the stability criteria established here for ndimensional linear DDEs. On the other hand, there are several recent works where explicit conditions for the exponential asymptotic stability of linear DDEs depending on the size of delays were found, see Berezansky and Braverman [2,3] for the scalar case and Berezansky et al [4,5] for n-dimensional systems, as well as references therein.…”

Integral conditions for nonuniform $μ$-dichotomy on the half-line

“…We refer the book [23] for the relation between (uniform) exponential dichotomies and Lyapunov functions. For stability results via Lyapunov functions in the context of delay equations, we refer the work of Hatvani and collaborators [11,12,16].…”