Existence of an optimal control for fractional stochastic partial neutral integro-differential equations with infinite delay

Abstract

In this paper we study optimal control problems governed by fractional stochastic partial neutral functional integro-differential equations with infinite delay in Hilbert spaces. We prove an existence result of mild solutions by using the fractional calculus, stochastic analysis theory, and fixed point theorems with the properties of analytic α-resolvent operators. Next, we derive the existence conditions of optimal pairs of these systems. Finally an example of a nonlinear fractional stochastic parabolic optimal control system is worked out in detail. c ⃝2015 All rights reserved.Keywords: Fractional stochastic partial neutral functional integro-differential equations, optimal controls, infinite delay, analytic α-resolvent operator, fixed point theorem.

The Solvability and Fractional Optimal Control for Semilinear Stochastic Systems

“…Li and Liu [11] presented the optimal control for nonlinear impulsive differential equations in Banach space setting. Yan and Lu [25] investigated the optimal control problems for fractional stochastic neutral integro-differential equations with infinite delay in a Hilbert space by using the fractional calculus, stochastic analysis theory, and fixed point theorem.…”

On a fractional impulsive partial stochastic integro-differential equation with state-dependent delay and optimal controls

“…Suganya et al [39] proved the existence results for an impulsive fractional neutral integro-differential equation with state-dependent delay and non-instantaneous impulses in Banach spaces. Yan and Lu [42] concerned with a class of fractional impulsive partial neutral stochastic integro-differential equations with not instantaneous impulses in Hilbert spaces.…”

On a new class of impulsive stochastic partial neutral integro-differential equations

“…Using the Banach contraction mapping principle, the existence, uniqueness, and continuous dependence results of mild solution fractional functional integro-differential equations with not instantaneous impulse has been reported in [20]. Yan and Lu [21] considered a class of fractional impulsive partial neutral stochastic integro-differential equations with not instantaneous impulses in Hilbert spaces. Yu and Wang [22] discussed the existence of mild solutions for periodic boundary value problems with not instantaneous impulse on Banach spaces using the Banach contraction mapping principle and Krasnoselskii's fixed point theorem.…”

“…Although the papers [17][18][19][20][21][22] studied the existence results for nonlinear impulsive differential and integro-differential equations with not instantaneous impulse, besides the fact that [17][18][19][20][21][22] applies to the existence of systems under the compactness assumptions or Lipschitz conditions, the class of nonlinear impulsive systems is also different from the one studied here. Further, stochastic systems with infinite delay and not instantaneous impulse deserve a study because they describe a kind of system present in the real world.…”

“…Specifically, we only concerned with the case in which R(·) is analytic and f, F satisfy the Carathéodory condition. The known results appeared in [17][18][19][20][21][22] are generalized to the impulsive stochastic systems settings and the case of infinite delay in the α-norm. Moreover, the results appear are also new for deterministic systems with not instantaneous impulse.…”