Optimal Control for Neutral Stochastic Integrodifferential Equations with Infinite Delay Driven by Poisson Jumps and Rosenblatt Process

Abstract

In this paper, we investigate the optimal control problems for a class of neutral stochastic integrodifferential equations (NSIDEs) with infinite delay driven by Poisson jumps and the Rosenblat process in Hilbert space involving concrete-fading memory-phase space, in which we define the advanced phase space for infinite delay for the stochastic process. First, we introduce conditions that ensure the existence and uniqueness of mild solutions using stochastic analysis theory, successive approximation, and Grimmer’s resolvent operator theory. Next, we prove exponential stability, which includes mean square exponential stability, and this especially includes the exponential stability of solutions and their maps. Following that, we discuss the existence requirements of an optimal pair of systems governed by stochastic partial integrodifferential equations with infinite delay. Then, we explore examples that illustrate the potential of the main result, mainly in the heat equation, filter system, traffic signal light systems, and the biological processes in the human body. We conclude with a numerical simulation of the system studied. This work is a unique combination of the theory with practical examples and a numerical simulation.

Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient

“…Time-fractional stochastic partial integrodifferential equations (TFSPIDEs) are related to diffusion or slow diffusion in materials with memory. (For connected deterministic PDEs, see [4][5][6]; for connected stochastic PDEs, see [7,8]; and, for the associated stochastic integral equations (SIEs), see [9][10][11].) Expanded upon by [11], Brownian-time processes (BTP) provide the foundation for the deterministic version of the TFSPIDEs.…”

“…4), we can obtain lim n→+∞ supt∈I time :I time ∈Q n |Θ β,d,x (t, 2 −n (log) −1 ) − Θ β,d,x (2 −n [t2 n ], 2 −n (log) −1 )| = 0 a.s.…”