Existence of relaxed optimal control for $G$-neutral stochastic functional differential equations with uncontrolled diffusion

In this paper, we study under refined Lipschitz hypothesis, the question of existence and uniqueness of solution of controlled neutral stochastic functional differential equations driven by G -Brownian motion ( G -NSFDEs in short). An existence of a relaxed optimal control where the neutral and diffusion terms do not depend on the control variable was the main result of the article. The latter is done by using tightness techniques and the weak convergence techniques for each probability measure in the set of all possible probabilities of our dynamic. A motivation of our work is presented and a numerical analysis for the uncontrolled G -NSFDE is given.