Abstract
A class of generalized impulsive stochastic functional differential systems with delayed impulses is considered. By employing piecewise continuous Lyapunov functions and the Razumikhin techniques, several criteria on the exponential stability and uniform stability in terms of two measures for the mentioned systems are obtained, which show that unstable stochastic functional differential systems may be stabilized by appropriate delayed impulses. Based on the stability results, delayed impulsive controllers which mean square exponentially stabilize linear stochastic delay systems are proposed. Finally, numerical examples are given to verify the effectiveness and advantages of our results.
Highlights
In recent years, the theory of impulsive functional differential systems (IFDSs) has attracted an increasing interest due to the wide existence of impulse effects and time delays in realworld systems
(3) the global exponential stability of the prescribed motion y(t), if h(t, x) = h0(t, x) = |x − y|; (4) the global exponential stability of the invariant set A ∈ Rn, if h(t, x) = h0(t, x) = d(x, A), where d(x, A) is the distance of x from the set A; (5) the global exponential orbital stability of a periodic solution, if h(t, x) = h0(t, x) = d(x, C), where C is the closed orbit in the phase space
An important special case of system (1) is the following ISFDSs-nDI, in which the state variables on impulses are not related to the time delay dx (t) = f (t, xt) dt + σ (t, xt) dB (t), t ⩾ t0, t ≠ tk, (31)
Summary
The theory of impulsive functional differential systems (IFDSs) has attracted an increasing interest due to the wide existence of impulse effects and time delays in realworld systems. By employing Lyapunov functions coupled with Razumikhin techniques, [9] investigated the asymptotic stability and practical stability for a class of generalized IFDSs-DI, while [10, 11] further established several criteria for the exponential stability of the systems. Applying the Lyapunov-Razumikhin techniques, [13] investigated both moment and almost sure exponential stability of impulsive stochastic functional differential systems with delayed impulses (ISFDSs-DI). In [14], the authors started the study of robust stability and state-feedback stabilization of uncertain impulsive stochastic delayed differential systems with linear delayed impulses. Motivated by the above discussion, the present paper will employ the Razumikhin techniques to investigate the problems of stability analysis and impulsive stabilization of ISFDSs-DI.
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