Abstract
In this paper, a class of time-space fractional stochastic delay control problems with fractional noises and Poisson jumps in a bounded domain is considered. The proper function spaces and assumptions are proposed to discuss the existence of mild solutions. In particular, approximate strategy is used to obtain the existence of mild solutions for the problem with linear fractional noises; fixed point theorem is used to achieve the existence of mild solutions for the problem with nonlinear fractional noises. Finally, the approximate controllability of the problems with linear and nonlinear fractional noises is proved by the property of mild solutions.
Highlights
We study a class of fully nonlocal stochastic control problems with delay in a bounded domain O ⊂ R N :
If y is a mild solution of problem (9), there exists a positive constant M such that kyk2CF ≤ M, t where M depends on α, T, H, M1, M2, M3, k φkCF, kvk L2 ([0,T ],V ) and kΦk1
If y is a mild solution of problem (1), there exists a positive constant Msuch that kyk2CF ≤ M, t where M depends on α, T, H, M1, M2, M4, M5, k φkCF, kvk L2 ([0,T ],V ), and kΦk1
Summary
We study a class of fully nonlocal stochastic control problems with delay in a bounded domain O ⊂ R N :. Fundamental solutions for fully nonlocal PDE problems were discussed in [19,20,21] These results encourage researchers to study the fractional stochastic partial differential equations (f-SPDE). Y(t) = y0 (t) ∈ X, t ∈ [−τ, 0], where A is a bounded abstract operator and X is a Hilbert space They obtained the existence and uniqueness of mild solutions. Li [23] discussed problem (2) with fractional time derivative, i.e., 0C Dtα y(t) instead of dy(t) in (2) and established the existence and uniqueness of mild solution by approximate method when indexes s ∈ Establish the existence and uniqueness of the mild solution to fully nonlocal stochastic delay control problems with both linear and nonlinear fractional noise by two different methods.
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