Indefinite Mean-Field Stochastic Linear-Quadratic Optimal Control: From Finite Horizon to Infinite Horizon
Indefinite Mean-Field Stochastic Linear-Quadratic Optimal Control: From Finite Horizon to Infinite Horizon
复制标题
不定平均场随机线性二次最优控制：从有限视野到无限视野
DOI：
10.1109/tac.2015.2509958
复制
发表时间：
2015-07
影响因子：
6.8
通讯作者：
Ni YH
中科院分区：
文献类型：
--
作者：
Ni Yuan-Hua;Ni Yuan-Hua;Zhang Ji-Feng;Li Xun;Ni YH
本文研究了有限地平线和无限地平线不定平均场随机线性二次最优控制问题。首先，介绍了有限水平问题的开环最优控制和闭环最优策略，并深入研究了它们的特征、区别和关系。开环最优控制可以定义为初始状态固定的开环最优控制，其存在性通过具有平稳条件和凸性条件的线性平均场正反向随机差分方程的可解性来表征。另一方面，证明了闭环最优策略的存在性等价于以下条件中的任意一个：一对广义差分Riccati方程的可解性，所有初始对的值函数的有限性，以及所有初始对的开环最优控制的存在性。然后证明了广义差分Riccati方程的解收敛于一对广义代数Riccati方程的解。通过对另一个广义代数Riccati方程的研究，得到了原方程的极大解的存在性以及稳定解为极大解的事实。最后，我们证明了用极大解来表示无限视界不定平均场线性二次最优控制的最优值。此外，对于最大解是否为稳定解的问题，给出了几种情况下的充分必要条件。
In this paper, the finite-horizon and the infinite-horizon indefinite mean-field stochastic linear-quadratic optimal control problems are studied. Firstly, the open-loop optimal control and the closed-loop optimal strategy for the finite-horizon problem are introduced, and their characterizations, difference and relationship are thoroughly investigated. The open-loop optimal control can be defined for a fixed initial state, whose existence is characterized via the solvability of a linear mean-field forward-backward stochastic difference equation with stationary conditions and a convexity condition. On the other hand, the existence of a closed-loop optimal strategy is shown to be equivalent to any one of the following conditions: the solvability of a couple of generalized difference Riccati equations, the finiteness of the value function for all the initial pairs, and the existence of the open-loop optimal control for all the initial pairs. It is then proved that the solution of the generalized difference Riccati equations converges to a solution of a couple of generalized algebraic Riccati equations. By studying another generalized algebraic Riccati equation, the existence of the maximal solution of the original ones is obtained together with the fact that the stabilizing solution is the maximal solution. Finally, we show that the maximal solution is employed to express the optimal value of the infinite-horizon indefinite mean-field linear-quadratic optimal control. Furthermore, for the question whether the maximal solution is the stabilizing solution, the necessary and the sufficient conditions are presented for several cases.
登录
查看更多内容
无限视野中的线性二次随机二人零和微分博弈
DOI：
10.1051/cocv/2015024
发表时间：
2016
期刊：
ESAIM: Control, Optimisation and Calculus of Variations
影响因子：
--
作者：
Sun Jingrui;Yong Jiongmin;Zhang Shuguang
通讯作者：
Zhang Shuguang
DOI：
10.1109/9.863597
发表时间：
2000-06
期刊：
IEEE Trans. Autom. Control.
影响因子：
--
作者：
M. A. Rami;X. Zhou
通讯作者：
M. A. Rami;X. Zhou
DOI：
--
发表时间：
1999-06
期刊：
--
影响因子：
--
作者：
J. Yong;X. Zhou
通讯作者：
J. Yong;X. Zhou
影响因子：
--
作者：
Y. Achdou;P. Cardaliaguet;François Delarue;A. Porretta;Filippo Santambrogio
通讯作者：
Y. Achdou;P. Cardaliaguet;François Delarue;A. Porretta;Filippo Santambrogio
影响因子：
1.8
作者：
Daniel Andersson;Boualem Djehiche
通讯作者：
Daniel Andersson;Boualem Djehiche