Stochastic time-optimal control for time-fractional Ginzburg-Landau equation with mixed fractional Brownian motion

A theoretical approach for solving time-fractional stochastic Ginzburg-Landau equation with mixed fractional Brownian motion in Hilbert space is elaborated. Initially, the stochastic partial differential system is reformulated in the Hilbert space by using the properties of fractional order space and fractional Laplacian. We establish the existence of mild solutions by employing Mittag-Leffler functions, stochastic analysis, and Krasnoselskii's fixed point theorem. A sufficient condition for the existence of a Lagrange optimal control problem is established via Balder's theorem. Further, the existence of stochastic time-optimal control and stochastic optimal time are analyzed for the proposed control system. An example is given to illustrate the developed theory. Finally, an application to the stochastic optimal control of hydropower plant model is provided. The optimal control is termed as the amount of release of water through the reservoir and it is controlled with a suitable performance index.