Optimal control of Sobolev‐type stochastic Hilfer fractional non‐instantaneous impulsive differential inclusion involving Poisson jumps and Clarke subdifferential

Abstract

This article is concerned with the optimal control of Sobolev-type Hilfer fractional non-instantaneous impulsive differential inclusion driven by Poisson jumps and Clarke subdifferential. Initially, the existence of a mild solution is established for the proposed Hilfer type fractional problem with novel ideas of non-instantaneous impulses. The non-linear alternative of Leray-Schauder type fixed point theorem, stochastic analysis, the measure of non-compactness and the multivalued analysis are applied to prove the mild solution. Further, the existence of optimal control is derived by employing Balder's theorem. Finally, the application as a stochastic dam pollution model is provided to illustrate the developed theoretical results.