On the linear-quadratic regulator problem in one-dimensional linear fractional stochastic systems

In this paper, for one-dimensional stochastic linear fractional systems in terms of the Riemann–Liouville fractional derivative, the optimal control is derived. It is assumed that the state is completely observable and all the information regarding this is available. The formulation leads to a set of stochastic fractional forward and backward equation in the Riemann–Liouville sense. The proposed method has been checked via some numerical simulations which show the effectiveness of the fractional stochastic optimal algorithm.