带混合项随机LQ问题在状态期望等式约束下的最优控制  

Optimal Control of General Stochastic Linear-quadratic Problems with State Expectation Equality Constraint

作　　者：姚楠 李继泽 周永辉 YAO Nan;LI Ji-zhe;ZHOU Yong-hui(School of Mathematical Sciences,Guizhou Normal University,Guiyang 550025,China;不详)

机构地区：[1]贵州师范大学数学科学学院,贵州贵阳550025 [2]贵州师范大学大数据与计算机科学学院,贵州贵阳550025

出　　处：《遵义师范学院学报》2025年第2期86-90,共5页Journal of Zunyi Normal University

基　　金：国家自然科学基金资助项目(11861025);贵州省科技厅项目(黔科中引地[2022]4055);贵州省教育厅自然科学研究项目(QJJ[2023]011)。

摘　　要：研究一类在期望等式约束下的随机LQ问题的最优控制,这里受控状态系统是线性的,其期望受到等式约束,而运行成本是关于状态和控制的混合项二次型函数。首先,引入两个随机Riccati方程的系统,将运行成本化简为仅含控制的标准二次型函数。其次,在该方程组的可解性和期望等式约束满射的条件下,运用拉格朗日乘子法,将原问题转化为等价的无约束含参数的随机最优控制问题。进而,引入对应的具有显示解的随机Riccati方程,得到该无约束含参数随机控制问题的最优控制。最后,应用凸优化的对偶性理论,得到原问题的最优控制。This paper studies optimal control of a stochastic linear-quadratic problem, where the controlled state is linear and constrainedby an equality on its expectation, and the running cost is a quadratic function with mixed terms of both the state and the control. Firstly,a systemof two stochastic Riccati equations is introduced to reduce the running cost to a standard quadratic function containing only thecontrol. Secondly, under the solvability and a surjective condition of constraint mapping for the expectation equality, the original problemistransformed into an equivalent unconstrained stochastic optimal control problemwith parameters by applying the Lagrange multipliermethod. Further, the corresponding stochastic Riccati equation with explicit solutions is introduced to obtain the optimal controlof this unconstrained parameter-containing stochastic control problem. Finally, the duality theory of convex optimization is applied toobtain the optimal control of the original problem.

关 键 词：带混合项随机LQ最优控制 状态期望等式约束 随机Riccati方程 拉格朗日对偶性理论 

分 类 号：O211[理学—概率论与数理统计]