Abstract
In this paper, we study the relative controllability of a fractional stochastic system with pure delay in finite dimensional stochastic spaces. A set of sufficient conditions is obtained for relative exact controllability using fixed point theory, fractional calculus (including fractional delayed linear operators and Grammian matrices) and local assumptions on nonlinear terms. Finally, an example is given to illustrate our theory.
Highlights
The integrals and derivatives of noninteger order and the fractional integro-differential equations arise in recent research in theoretical physics, mechanics and applied mathematics and fractional calculus is an effective tool to explain bodily structures that have long-term reminiscence and lengthy-range spatial integration
We extend the representation of the solution introduced in [26] for fractional linear systems to nonlinear stochastic systems and present relatively exact controllability results for the following stochastic systems: CD−q τ+ CD−q τ+ y (t) + A2y(t − τ )
We investigate the relative controllability of the fractional stochastic system with pure delay
Summary
The integrals and derivatives of noninteger order and the fractional integro-differential equations arise in recent research in theoretical physics, mechanics and applied mathematics and fractional calculus is an effective tool to explain bodily structures that have long-term reminiscence and lengthy-range spatial integration (see [1, 14, 24]). A solution representation and relative controllability results for higher-order linear discrete delayed systems with a single delay using a special matrix functions called discrete delayed sine and cosine matrices can be found in [5]. Sufficient conditions are established for controllability of second-order nonlinear stochastic delay systems using fixed point theory, delayed sine and cosine matrices and delayed Grammian matrices in [31]. In [26] the representation of solution for the Cauchy problem (1) is presented, but it is necessary to analyze the relatively exact controllability of nonlinear stochastic systems with pure delay. We establish necessary and sufficient conditions for linear stochastic systems using controllability Grammian matrices and linear operators, which are defined by delayed fractional cosine and sine matrices, and the minimum energy control problem. We give a solution representation for the inhomogeneous stochastic system, and we define the delayed Grammian matrix using fractional delayed sine and cosine matrices
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