Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations

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Abstract

Recently, we initiated in [Systems Control Lett., 26 (1995), pp. 245-251] the study of exponential stability of neutral stochastic functional differential equations, and in this paper, we shall further our study in this area. We should emphasize that the main technique employed in this paper is the well-known Razumikhin argument and is completely different from those used in our previous paper [Systems Control Lett., 26 (1995), pp. 245-251]. The results obtained in [Systems Control Lett., 26 (1995), pp. 245-251] can only be applied to a certain class of neutral stochastic functional differential equations excluding neutral stochastic differential delay equations, but the results obtained in this paper are more general, and they especially can be used to deal with neutral stochastic differential delay equations. Moreover, in [Systems Control Lett., 26 (1995), pp. 245-251], we only studied the exponential stability in mean square, but in this paper, we shall also study the almost sure exponential stability. It should be pointed out that although the results established in this paper are applicable to more general neutral-type equation, for a particular type of equation discussed in [Systems Control Lett., 26 (1995), pp. 245-251], the results there are sharper.

LanguageEnglish
Pages389-401
Number of pages13
JournalSIAM Journal on Mathematical Analysis
Volume28
Issue number2
Publication statusPublished - Mar 1997

Fingerprint

Stochastic Functional Differential Equations
Neutral Functional Differential Equation
Exponential Stability
Asymptotic stability
Differential equations
Control systems
Stochastic Differential Delay Equations
Theorem
Almost Sure Exponential Stability
Neutral Type
Mean Square

Keywords

  • Borel-Cantelli lemma
  • Brownian motion
  • Doob martingale inequality
  • exponential stability
  • Razumikhin-type theorem

Cite this

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title = "Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations",
abstract = "Recently, we initiated in [Systems Control Lett., 26 (1995), pp. 245-251] the study of exponential stability of neutral stochastic functional differential equations, and in this paper, we shall further our study in this area. We should emphasize that the main technique employed in this paper is the well-known Razumikhin argument and is completely different from those used in our previous paper [Systems Control Lett., 26 (1995), pp. 245-251]. The results obtained in [Systems Control Lett., 26 (1995), pp. 245-251] can only be applied to a certain class of neutral stochastic functional differential equations excluding neutral stochastic differential delay equations, but the results obtained in this paper are more general, and they especially can be used to deal with neutral stochastic differential delay equations. Moreover, in [Systems Control Lett., 26 (1995), pp. 245-251], we only studied the exponential stability in mean square, but in this paper, we shall also study the almost sure exponential stability. It should be pointed out that although the results established in this paper are applicable to more general neutral-type equation, for a particular type of equation discussed in [Systems Control Lett., 26 (1995), pp. 245-251], the results there are sharper.",
keywords = "Borel-Cantelli lemma, Brownian motion, Doob martingale inequality, exponential stability, Razumikhin-type theorem",
author = "Xuerong Mao",
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T1 - Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations

AU - Mao, Xuerong

N1 - (c) 1997 Society for Industrial and Applied Mathematics

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N2 - Recently, we initiated in [Systems Control Lett., 26 (1995), pp. 245-251] the study of exponential stability of neutral stochastic functional differential equations, and in this paper, we shall further our study in this area. We should emphasize that the main technique employed in this paper is the well-known Razumikhin argument and is completely different from those used in our previous paper [Systems Control Lett., 26 (1995), pp. 245-251]. The results obtained in [Systems Control Lett., 26 (1995), pp. 245-251] can only be applied to a certain class of neutral stochastic functional differential equations excluding neutral stochastic differential delay equations, but the results obtained in this paper are more general, and they especially can be used to deal with neutral stochastic differential delay equations. Moreover, in [Systems Control Lett., 26 (1995), pp. 245-251], we only studied the exponential stability in mean square, but in this paper, we shall also study the almost sure exponential stability. It should be pointed out that although the results established in this paper are applicable to more general neutral-type equation, for a particular type of equation discussed in [Systems Control Lett., 26 (1995), pp. 245-251], the results there are sharper.

AB - Recently, we initiated in [Systems Control Lett., 26 (1995), pp. 245-251] the study of exponential stability of neutral stochastic functional differential equations, and in this paper, we shall further our study in this area. We should emphasize that the main technique employed in this paper is the well-known Razumikhin argument and is completely different from those used in our previous paper [Systems Control Lett., 26 (1995), pp. 245-251]. The results obtained in [Systems Control Lett., 26 (1995), pp. 245-251] can only be applied to a certain class of neutral stochastic functional differential equations excluding neutral stochastic differential delay equations, but the results obtained in this paper are more general, and they especially can be used to deal with neutral stochastic differential delay equations. Moreover, in [Systems Control Lett., 26 (1995), pp. 245-251], we only studied the exponential stability in mean square, but in this paper, we shall also study the almost sure exponential stability. It should be pointed out that although the results established in this paper are applicable to more general neutral-type equation, for a particular type of equation discussed in [Systems Control Lett., 26 (1995), pp. 245-251], the results there are sharper.

KW - Borel-Cantelli lemma

KW - Brownian motion

KW - Doob martingale inequality

KW - exponential stability

KW - Razumikhin-type theorem

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