Boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and Poisson jumps

M Elsaid Ramadan,Hamdy M Ahmed,Mahmoud M El-Borai

doi:10.1186/s13662-019-2028-1

Abstract

By using stochastic analysis, fractional analysis, compact semigroups and the Schauder fixed-point theorem, we discuss the approximate boundary controllability of a nonlocal Hilfer fractional stochastic differential system with fractional Brownian motion and a Poisson jump. In addition, we establish the sufficient conditions for exact null controllability for the same problem. Finally, an example is given to illustrate the results obtained.

Highlights

1 Introduction Fractional calculus has been applied to the description of problems that arise in a variety of fields, including finance, physics, geomagnetics, thermodynamics, and optimal control
We in this paper investigate the sufficient conditions for approximate boundary controllability and null boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and Poisson jump in the following form:
3 Approximate boundary controllability we discuss the approximate controllability for the system (1.1), so we introduce the following linear Hilfer fractional differential system with control on the boundary:

Summary

Introduction

Fractional calculus has been applied to the description of problems that arise in a variety of fields, including finance, physics, geomagnetics, thermodynamics, and optimal control. Suppose {ω(t)}t≥0 is a Wiener process defined on (Ω, F, {Ft}t≥0, P) with values in the Hilbert space K and {BH (t)}t≥0 is a fractional Brownian motion (fBm) with Hurst parameter. In order to define Wiener integrals with respect to the Q-fBm, we introduce the space L02 := L02(Y , X) of all Q-Hilbert Schmidt operators ψ : Y → X. (H8) The function h : J × X × V → X satisfies the following two conditions: (i) The function h : J × X × V → X is continuous; (ii) for each positive number r ∈ N , there is a positive function χr(·) : J → R+ such that sup E h(t, x, v) 2λ(dv) ≤ χr(t), x 2≤r V the function s → (t – s)μ–1χr(s) ∈ L1([0, t], R+), and there exists a δ4 > 0 such that lim inf r→∞. + Pμ(t – s) h s, x(s), v N (ds, dv), t ∈ J

L02 ds

Applications