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

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.