Razumikhin‐type theorems on general decay stability and robustness for stochastic functional differential equations

Abstract

This paper establishes Razumikhin-type theorems on general decay stability for stochastic functional differential equations. This improves existing stochastic Razumikhin-type theorems and can make us examine the stability with general decay rate in the sense of the pth moment and almost sure. These stabilities may be specialized as the exponential stability and the polynomial stability. When the almost sure stability is examined, the conditions of this paper may defy the linear growth condition for the drift term, which implies that the theorems of this paper can work for some cases to which the existing results cannot be applied. This paper also examines some sufficient criteria under which this stability is robust. To illustrate applications of our results clearly, this paper also gives two examples and examines the exponential stability and the polynomial stability, respectively.By condition (H2), letting u = /( +¯ ) and applying the inequality (40) yield (here we assume that ,¯ = 0, otherwise this computation is easier) |g(t, )| 2 1 (t) ( +¯ )| (0)| 2 +¯ ( +¯ ) 0 − | ( )| 2 d ( ) .

Synchronization of a Class of Switched Neural Networks with Time-Varying Delays via Nonlinear Feedback Control

“…Definition 4 [54]: Function ψ : R + → (0, ∞) is said to be ψ-type function if this function satisfies the following conditions.…”

“…Definition 5 [54]: The error system (7) is said to be ψ-type stable if there exists a constant γ > 0 such that…”

Divided State Feedback Control of Stochastic Systems

“…Remark 3: In the previous literature, the Lyapunov function method is always accompanied with Razumikhin technique, see [23], [26], [27], and [30]. If we employ this technique, we will continue the derivation from the obtained estimation (17) in the Appendix for the proof of Theorem 1.…”

Razumikhin-Type Theorems on p-th Moment Stability for Stochastic Switching Nonlinear Systems with Delay

“…Later, this technique was appropriately developed and extended to some other stochastic systems, such as hybrid stochastic delay interval systems [9] and impulsive stochastic delay differential systems [10]. Recently, some researchers have introduced ψ -type function and extended the stability results to γ ψ stability, including the exponentialstability as a special case in [11] [12]. In [13], the researchers utilize multiple Lyapunov functions investigate the stability of stochastic switching nonlinear systems.…”