Approximate Controllability of Impulsive Stochastic Evolution Equations

Abstract

Abstract. In this paper, we study the approximate controllability of nonlinear impulsive stochastic di¤erential equations in a real separable Hilbert space. We prove the approximate controllability of nonlinear impulsive stochastic control systems under the assumption that the corresponding linear system is approximately controllable. By using the stochastic analysis theory and a fixed point strategy, su‰cient conditions are formulated and proved. Moreover, an example is also provided to illustrate the e¤ectiveness of the result.

On the approximate controllability of semilinear fractional differential systems

“…Therefore, it is important, in fact necessary to study the weaker concept of controllability, namely approximate controllability for nonlinear systems. In the recent literature, there have been few papers on the approximate controllability of the nonlinear evolution systems under different conditions [9,[12][13][14][15]. Fu and Mei [16] investigated the approximate controllability of semilinear neutral functions differential systems with finite delay.…”

Existence and Controllability Results for Fractional Stochastic Semilinear Differential Inclusions

“…[8,9,19,21]. The controllability of nonlinear stochastic systems in infinite dimensional spaces has been extensively investigated by several authors; see [20] and the references therein. Among them, Balachandran and Dauer [3] investigated the controllability of nonlinear systems in Banach spaces.…”

“…Very recently, Debbouche and Baleanu [6] derived a set conditions for the controllability of a class of fractional evolution nonlocal impulsive quasilinear delay integrodifferential systems by using the theory of fractional calculus and fixed point technique. The approximate controllability problems for infinite-dimensional dynamical systems has been studied in [9,[18][19][20]. Sakthivel et al [18] studied the approximate controllability of nonlinear deterministic and stochastic evolution systems with unbounded delay in abstract spaces.…”

Approximate controllability of abstract stochastic impulsive systems with multiple time‐varying delays

“…The motivations of this paper are approximate controllability that proves to be more practical compared with complete controllability and stochastic impulsive systems, with and without delays, that serve as an abstract formulation for many control systems described by partial or functional differential equations. However, to the best of our knowledge, only few results are available on the approximate controllability of stochastic impulsive systems, except and , let alone the research about the more general stochastic semilinear impulsive systems with time‐varying delays. And Sakthivel and Subalakshmi and Balachandran discussed, respectively, the approximate controllability of stochastic impulsive systems and stochastic impulsive integro‐differential systems.…”

“…However, to the best of our knowledge, only few results are available on the approximate controllability of stochastic impulsive systems, except and , let alone the research about the more general stochastic semilinear impulsive systems with time‐varying delays. And Sakthivel and Subalakshmi and Balachandran discussed, respectively, the approximate controllability of stochastic impulsive systems and stochastic impulsive integro‐differential systems.…”