Optimal control of stochastic multi-valued impulsive non-autonomous differential equations with delayed force term: optimal mild solution

Abstract This article is concerned with the optimal mild solution and optimal control for stochastic Caputo fractional multi-valued impulsive non-autonomous differential equations with delayed force term and fractional Brownian motion in Hilbert spaces. We establish the existence of mild solution and optimal mild solution for the considered system by employing the Bohnenblust-Karlin fixed point theorem in the absence of Lipschitz conditions. Next, the optimal control of the considered non-autonomous differential system is derived with the aid of Balder’s theorem. To demonstrate the applicability of the theoretical results, we present a concrete model of ship motion and control, formulated as a stochastic Caputo fractional multi-valued impulsive non-autonomous system with delayed hydrodynamic forces, environmental randomness and impulsive events such as docking or collision. The associated cost functional balances trajectory tracking with control effort, ensuring safe and fuel efficient operation. Furthermore, an example is given to support the theoretical results.