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

Abstract

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.

Keywords:

• Existence of mild solution
• fractional Brownian motion
• Ginzburg–Landau equation
• stochastic optimal control
• time-optimal control

2020 Mathematics Subject Classification:

• 35Q56
• 35R11
• 49J20
• 60G57
• 60H15

Acknowledgments

The authors of this paper should like to thank the Editors and Associate Editors of this journal as well as the anonymous reviewers who have generously given up valuable time to review. The success of this work depends upon their care and competence. Their conscientiousness is much appreciated.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by Council of Scientific and Industrial Research (CSIR), Govt. of India under EMR Project, F.No: 25(0273)/17/EMR-II dated 27-04-2017, and in part by NSF of China (Nos. 11671142 and 11771075), Science and Technology Commission of Shanghai Municipality (STCSM) (Grant No. 18dz2271000).