Qualitative study on Sobolev-type delayed fractional stochastic impulsive system: existence and controllability
O. P. Sharma, Ramesh Kumar Vats
Abstract
The aim of this research is to study the sufficient conditions for the existence of the mild solution and approximate controllability results of a new class of the Caputo conformable fractional Sobolev-type delayed stochastic system with non-instantaneous impulses in a separable Hilbert space. Since, the conformable fractional derivative retains several classical properties such as the mean value theorem, Rolle’s theorem, the product and quotient rules, and linearity, which distinguish it from traditional fractional derivatives, including the Riemann-Liouville, Caputo, and Hilfer fractional derivatives. Therefore, the conformable fractional derivative is simpler and faster but ignores history, while the Caputo conformable fractional derivative offers a balance capturing some memory with easier calculations. Firstly, the Riemann-Liouville conformable fractional integral operator is used to convert the proposed system into an equivalent fixed-point problem. Then, the Schauder’s fixed point theorem is employed to derive the existence result. Further, the approximate controllability result of the proposed system is established under the consideration that the corresponding linear system is approximately controllable. The main tools applied in this study are fractional calculus, semigroups of bounded linear operators, stochastic analysis and fixed point approach. At the end of the paper, a concrete example is provided to validate the theoretical results.