Abstract. In the G-framework, we establish existence of an optimal stochastic relaxed control for stochastic differential equations driven by a G-Brownian motion.

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In this paper, we study the question of existence and uniqueness of solution of neutral stochastic functional differential equations driven by G-Brownian motion (G-NSFDEs in short) on Banach space driven by relaxed controls in which the neutral term and diffusion do not depend on the control variable. By using tightness techniques and the weak convergence techniques for each probability measure in the set of all possible probabilities of our dynamic, we prove the existence of an optimal relaxed control. A motivation of are work is presented and a Numerical analysis for the uncontrolled G-NSFDE is given.