Relatively exact controllability of fractional stochastic delay system driven by Lévy noise

Abstract

In this article, we consider the relatively exact controllability of fractional stochastic delay system (FSDS) driven by Lévy noise. First, we derive the solution of linear FSDS via delayed matrix functions of Mittag-Leffler (M-L). Subsequently, by virtue of the controllability Grammian matrix, we explore the relatively exact controllability of linear FSDS. In addition, with the aid of Jensen's inequality, Hölder's inequality, and Itô's isometry, the existence and uniqueness of the considered nonlinear FSDS are investigated by employing Banach contraction principle. Thereafter, the relatively exact controllability of nonlinear FSDS is discussed. Finally, the theoretical results are supported through examples.

Relatively exact controllability of fractional stochastic neutral system with two incommensurate constant delays

“…Thus, we have which implies that all the conditions in Theorem 1 are satisfied. So system (18) is relatively exactly controllable.…”

“…Wang et al [19] considered the relatively exact controllability by controllability Grammian matrices and linear operators, which are defined delayed fractional cosine and sine matrices. In previous studies [18,19], the authors only analyze single delay systems and the Mittag-Leffler matrixs are undergeneralized.…”

“…In fact, the relatively exact controllability [17] means the system can reach the expected exact value under the action of the control function. In the work of Huang and Luo [18], the authors study the relatively exactly controllability problems of fractional stochastic single delay system by delayed Mittag–Leffler matrix function. Wang et al [19] considered the relatively exact controllability by controllability Grammian matrices and linear operators, which are defined delayed fractional cosine and sine matrices.…”