Global mild solutions and total controllability of fuzzy conformable fractional stochastic delay systems with non-instantaneous impulses and Poisson jumps

This manuscript is concerned with the solvability and total controllability of the fuzzy stochastic delay differential system with non-instantaneous impulses and Poisson jumps under the conformable fractional derivative, which is new to the literature. Such systems naturally arise in real-world processes where uncertainty, memory effects, and abrupt dynamic changes occur simultaneously. The existence and uniqueness results are obtained for the proposed system using the Banach contraction principle. Next, the global solution is derived with the help of generalised Gronwall’s inequality. Furthermore, the total controllability results are established for the presented fuzzy system. A new class of control functions is introduced to regulate the system at the termination of time intervals and on each impulsive event, while incorporating stochastic disturbances. This approach yields to comprehensive controllability outcomes, often termed as total controllability. In support, an example is given to validate the obtained theoretical outcomes. Furthermore, numerical simulations are presented to illustrate the presented model. The technology under discussion will also be used in a number of practical applications, such as traffic flow regulation, population dynamics, climate-driven environmental processes, communication networks, and medical treatment response systems.