具终端状态约束的无穷维随机发展方程的线性二次最优控制

Linear quadratic optimal control problem for stochastic evolution equations with terminal state constraints in infinite dimensions

1968年,Wonham提出随机线性二次最优控制问题.随后,1976年Bismut开始研究带有随机系数的随机线性二次最优控制问题.直到1998年,陈、李和周首次成功解决了具有不定控制加权项的随机系数的随机线性二次最优控制问题.此后,越来越多的研究者开始对随机线性二次最优控制问题产生兴趣.近二十年来,人们逐渐开始研究以无穷维随机发展方程为控制系统的线性二次最优控制问题.另一方面,实际应用中的控制系统的状态变量往往需要满足一些约束条件.在此背景下,本文研究了具终端状态约束的随机发展方程的线性二次最优控制问题.基于算子值Riccati方程可解性、控制系统的适当能控性及拉格朗日对偶理论,本文得到了该约束问题最优控制的表达式.

In 1968,Wonham proposed the stochastic linear quadratic optimal control problem.Subsequently,in 1976 Bismut began to study the stochastic linear quadratic optimal control problems with random coefficients.Until 1998,Chen,Li,and Zhou successfully solved the stochastic linear quadratic optimal control problem with indefinite control weight costs for the first time.Since then,more and more researchers have become interested in stochastic linear quadratic optimal control problems.In the past two decades,people have gradually begun to study the linear quadratic optimal control problem of infinite dimensional stochastic evolution equations as control systems.On the other hand,the state variables of control systems in practical applications often need to meet some constraint conditions.In this context,we investigate the linear quadratic optimal control problem of stochastic evolution equations with terminal state constrains.Based on the solvability of operator-valued Riccati equations,appropriate controllability of control systems and Lagrangian duality theory,we obtain the expression for the optimal control of the constrained problem.