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.

摘要

阐述了在希尔伯特空间中求解具有混合分数布朗运动的时间分数随机 Ginzburg-Landau 方程的理论方法。最初，利用分数阶空间和分数拉普拉斯算子的性质，在希尔伯特空间中重构随机偏微分系统。我们通过使用 Mittag-Leffler 函数、随机分析和 Krasnoselskii 不动点定理来确定温和解的存在性。通过巴尔德定理建立了拉格朗日最优控制问题存在的充分条件。此外，针对所提出的控制系统，分析了随机时间最优控制和随机最优时间的存在性。给出一个例子来说明发展的理论。最后，提供了在水电站模型随机优化控制中的应用。最佳控制被称为通过水库的泄水量，它由合适的性能指标控制。