论文标题
随机系数的无限维度中随机线性二次控制问题的最佳反馈控制
Optimal Feedback Controls of Stochastic Linear Quadratic Control Problems in Infinite Dimensions with Random Coefficients
论文作者
论文摘要
表征与随机系数随机演变方程的一般线性二次最佳控制问题的最佳反馈控制的长期未解决的问题。在[21]的一些假设中给出了解决该问题的解决方案，这些假设可以用于有趣的混凝土模型，例如受控的随机波动方程，受控的随机schrödinger方程等。更确切地说，作者确定了最佳反馈操作员的存在与相应操作员可溶解度之间的等价性，而相应的操作员valagety-backatostation satoctstoctications stochodation sttochodict riccati riccati riccati riccati riccati riccati riccati riccati riccati。但是，它们的结果不能涵盖某些重要的随机部分微分方程，例如随机热方程，随机的Stokes方程等。当前工作的关键贡献是放宽[21]中无绑定的线性算子$ $ a $的$ C_0 $组假设，并使用缩写半族假设。因此，我们的结果可能非常适用于随机抛物线偏微分方程的线性二次问题。为此，我们在上述的Riccati方程中引入了一个合适的概念，以及在有限维度的情况下甚至是一些新的技术。
It is a longstanding unsolved problem to characterize the optimal feedback controls for general linear quadratic optimal control problem of stochastic evolution equation with random coefficients. A solution to this problem is given in [21] under some assumptions which can be verified for interesting concrete models, such as controlled stochastic wave equations, controlled stochastic Schrödinger equations, etc. More precisely, the authors establish the equivalence between the existence of optimal feedback operator and the solvability of the corresponding operator-valued, backward stochastic Riccati equations. However, their result cannot cover some important stochastic partial differential equations, such as stochastic heat equations, stochastic stokes equations, etc. A key contribution of the current work is to relax the $C_0$-group assumption of unbounded linear operator $A$ in [21] and using contraction semigroup assumption instead. Therefore, our result can be well applicable in the linear quadratic problem of stochastic parabolic partial differential equations. To this end, we introduce a suitable notion to the aforementioned Riccati equation, and some delicate techniques which are even new in the finite dimensional case.
