An asymptotic stability and a uniform asymptotic stability for functional-differential equations

Ko, Youn-Hee

doi:10.1090/s0002-9939-1993-1169036-6

Cited by 2 publications

(2 citation statements)

References 11 publications

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“…). These properties were used by other authors [1,2,8,14,15,16,18,19,23,24,20,25,26]. Parts B in Theorems 2.2 and 2.3 show that these conditions can be essentially weakened if the Lyapunov functional is decrescent in Krasovskiȋ's sense.…”

Section: Asymptotic Stability For Nonautonomous Fde's 3561mentioning

confidence: 98%

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

Hatvani

2002

Trans. Amer. Math. Soc.

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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.

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Section: Asymptotic Stability For Nonautonomous Fde's 3561mentioning

confidence: 98%

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

Hatvani

2002

Trans. Amer. Math. Soc.

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“…At first this topic can be found in DRIVER [13] who reformulated, improved and extended much of the works of KRASOVSKIi and RAZUMIKHIN. Many other authors have made contributions by using Liapunov functionals and for some of the existing results we refer to [10,37,38,45,52].…”

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confidence: 99%

Families of Liapunov-Krasovskiį functionals and stability for functional differential equations

Athanassov

1999

Annali di Matematica pura ed applicata

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Among the methods for studying stability and related concepts for ordinary differential equations, Liapunov's second (or direct) method has become the most widely used. It is characterized by the use of certain (originally real-valued) auxiliary functions, now usually called Liapunov functions. Because of the difficulties in constructing suitable Liapunov functions, many authors have investigated various kinds of variations of Liapunov's second method, and new approaches to stability problems have emerged. One of the important approaches is that of considering finite or infinite families of continuous or discontinuous Liapunov functions. Such an approach leads to the generalization and improvement of existing results and proof, as well as finding new extensions and opening up of new fields (cf. [1,2], and the literature there).(*) Entrata in Redazione il 25 settembre 1997.

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An asymptotic stability and a uniform asymptotic stability for functional-differential equations

Abstract: Abstract. We consider a system of functional differential equation x'{t) = F{t, xt) and obtain conditions on a Liapunov functional to ensure the asymptotic stability and the uniform asymptotic stability of the zero solution.

Cited by 2 publications

References 11 publications

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

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

Families of Liapunov-Krasovskiį functionals and stability for functional differential equations

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