南方科技大学知识苑(SUSTech KC): MEAN-FIELD LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS WITH PARTIAL OBSERVATION

题名	MEAN-FIELD LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS WITH PARTIAL OBSERVATION
其他题名	部分观测的平均场线性二次最优控制问题
姓名	欧钰颖
姓名拼音	OU Yuying
学号	12032004
学位类型	硕士
学位专业	070105 运筹学与控制论
学科门类/专业学位类别	07 理学
导师	孙景瑞
导师单位	数学系
论文答辩日期	2022-05-08
论文提交日期	2022-06-22
学位授予单位	南方科技大学
学位授予地点	深圳
摘要	This paper is concerned with a class of mean-field linear-quadratic stochastic control problems for partially observed linear dynamical systems. And the control is required to be adopted to the filtration generated by the partially observation process. Since the filtration is not fixed, it is depending on the control, the linear structure of the admissible control set is thereby corrupted. The variation method fails in this case, and the orthogonal decomposition is applied to overcome this difffficulty. The state process is divided into two parts by filtering techniques: the first part X1 is a function of the control and the filtering process; the second part X2 is independent of the choice of the control. We construct the standard Brownian motion adapted to the filtering process and use the Brownian motion to obtain the stochastic differential equation of X1. The cost functional is similarly divided into two parts: the first part J1 is a function of X1 and the admissible control process; the second part J2 is a function of X2, independent of the choice of control. Thus, solving the optimal control is equivalent to solving the optimal control corresponding to J1.A feedback representation process is obtained for the optimal control via filtering. The optimal value is also obtained explicitly.
其他摘要	本文关注一类部分观测线性动力系统的平均场线性二次随机控制问题，并且要求控制适应于部分观测过程产生的滤波。由于滤波不是固定的，它依赖于控制，因此允许控制集的线性结构被破坏，变分法在这种情况下失效，文章采用正交分解来克服这个困难。通过滤波技术将状态过程分为正交的两部分：第一部分是控制和滤波过程的函数 X1，第二部分X2与选择的控制无关。我们构造适应于滤波过程的标准布朗运动，并且用布朗运动得到X1的随机微分方程。代价泛函也可类似地写为两部分：第一部分J1是X1和控制过程的函数，另一部分J2是X2 的函数，与控制的选择无关。因此，求解最优控制等价于求解J1对应的最优控制。最后，通过滤波得到最优控制的反馈形式，最优值也是明确得到的。
关键词	Mean-field Linear-quadratic Optimal Control Partially Observation Process Filtering Process
其他关键词	平均场线性二次最优控制部分观测过程滤波过程
语种	英语
培养类别	独立培养
入学年份	2020
学位授予年份	2022-07
参考文献列表	[1] KALMAN, EMIL R. Contributions to the theory of optimal control[J]. Bol. soc. mat. mexicana,1960, 5(2): 102-119. [2] KUSHNER H. Optimal stochastic control[J/OL]. IRE Transactions on Automatic Control,1962, 7(5): 120-122. DOI: 10.1109/TAC.1962.1105490. [3] WONHAM W M. On a matrix riccati equation of stochastic control[J]. SIAM Journal on Control, 1968, 6(4): 681-697. [4] YONG J. Linear-quadratic optimal control problems for mean-field stochastic differential equations[J]. SIAM journal on Control and Optimization, 2013, 51(4): 2809-2838. [5] BISMUT J M. Linear quadratic optimal stochastic control with random coefficients[J]. SIAM Journal on Control and Optimization, 1976, 14(3): 419-444. [6] BISMUT J M. Intégrales convexes et probabilités[J]. Journal of Mathematical Analysis and Applications, 1973, 42(3): 639-673. [7] BISMUT J M. Théorie probabiliste du contrôle des diffusions: volume 167[M]. American Mathematical Soc., 1976. [8] BISMUT J M. An introductory approach to duality in optimal stochastic control[J]. SIAM review, 1978, 20(1): 62-78. [9] PENG S. A general stochastic maximum principle for optimal control problems[J]. SIAM Journal on control and optimization, 1990, 28(4): 966-979. [10] CHEN S, LI X, ZHOU X Y. Stochastic linear quadratic regulators with indefinite control weight costs[J]. SIAM Journal on Control and Optimization, 1998, 36(5): 1685-1702. [11] CHEN S, ZHOU X Y. Stochastic linear quadratic regulators with indefinite control weight costs. ii[J]. SIAM Journal on Control and Optimization, 2000, 39(4): 1065-1081. [12] RAMI M A, MOORE J B, ZHOU X Y. Indefinite stochastic linear quadratic control and generalized differential riccati equation[J]. SIAM Journal on Control and Optimization, 2002, 40 (4): 1296-1311. [13] YONG J, ZHOU X Y. Stochastic controls: Hamiltonian systems and hjb equations: volume 43 [M]. Springer Science & Business Media, 1999. [14] CHEN S, YONG J. Stochastic linear quadratic optimal control problems[J]. Applied Mathematics and Optimization, 2001, 43(1): 21-45. [15] ZHOU X Y, LI D. Continuous-time mean-variance portfolio selection: A stochastic lq framework[J]. Applied Mathematics and Optimization, 2000, 42(1): 19-33. [16] SUN J, YONG J. Linear quadratic stochastic differential games: open-loop and closed-loop saddle points[J]. SIAM Journal on Control and Optimization, 2014, 52(6): 4082-4121. [17] KAC M. Foundations of kinetic theory[C]//Proceedings of The third Berkeley symposium on mathematical statistics and probability: volume 3. 1956: 171-197. [18] ANDERSSON D, DJEHICHE B. A maximum principle for sdes of mean-field type[J]. Applied Mathematics & Optimization, 2011, 63(3): 341-356. [19] BUCKDAHN R, DJEHICHE B, LI J. A general stochastic maximum principle for sdes of mean-field type[J]. Applied Mathematics & Optimization, 2011, 64(2): 197-216. [20] LI X, SUN J, YONG J. Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability[J]. Probability, Uncertainty and Quantitative Risk, 2016, 1(1): 1-24. [21] SUN J. Mean-field stochastic linear quadratic optimal control problems: Open-loop solvabilities [J]. ESAIM: Control, Optimisation and Calculus of Variations, 2017, 23(3): 1099-1127. [22] YONG J. Linear-quadratic optimal control problems for mean-field stochastic differential equations—time-consistent solutions[J]. Transactions of the American Mathematical Society, 2017, 369(8): 5467-5523. [23] LI X, SUN J, XIONG J. Linear quadratic optimal control problems for mean-field backward stochastic differential equations[J]. Applied Mathematics & Optimization, 2019, 80(1): 223-250. [24] AHMED N. Nonlinear diffusion governed by mckean–vlasov equation on hilbert space and optimal control[J]. SIAM journal on control and optimization, 2007, 46(1): 356-378. [25] BJORK T, MURGOCI A. A general theory of markovian time inconsistent stochastic control problems[J]. Available at SSRN 1694759, 2010. [26] MEYER-BRANDIS T, ØKSENDAL B, ZHOU X Y. A mean-field stochastic maximum principle via malliavin calculus[J]. Stochastics An International Journal of Probability and Stochastic Processes, 2012, 84(5-6): 643-666. [27] NI Y H, LI X, ZHANG J F. Indefinite mean-field stochastic linear-quadratic optimal control: from finite horizon to infinite horizon[J]. IEEE Transactions on Automatic Control, 2015, 61 (11): 3269-3284. [28] NI Y H, ELLIOTT R, LI X. Discrete-time mean-field stochastic linear–quadratic optimal control problems, ii: Infinite horizon case[J]. Automatica, 2015, 57: 65-77. [29] ELLIOTT R, LI X, NI Y H. Discrete time mean-field stochastic linear-quadratic optimal control problems[J]. Automatica, 2013, 49(11): 3222-3233. [30] SUN J, YONG J. Stochastic linear-quadratic optimal control theory: Differential games and mean-field problems[M]. Springer Nature, 2020. [31] LI X, TANG S. General necessary conditions for partially observed optimal stochastic controls [J]. Journal of applied probability, 1995, 32(4): 1118-1137. [32] HUANG J, WANG G, XIONG J. A maximum principle for partial information backward stochastic control problems with applications[J]. SIAM journal on Control and Optimization, 2009, 48(4): 2106-2117. [33] SHI J, WU Z. Maximum principle for partially-observed optimal control of fully-coupled forward-backward stochastic systems[J]. Journal of optimization theory and applications, 2010, 145(3): 543-578. [34] WANG G, WU Z, XIONG J. A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information[J]. IEEE Transactions on Automatic Control, 2015, 60(11): 2904-2916. [35] MA H, LIU B. Linear-quadratic optimal control problem for partially observed forward-backward stochastic differential equations of mean-field type[J]. Asian Journal of Control, 2016, 18(6): 2146-2157. [36] HUANG P, WANG G, ZHANG H. A partial information linear-quadratic optimal control problem of backward stochastic differential equation with its applications[J]. Science China Information Sciences, 2020, 63(9): 1-13. [37] WANG G, WANG W, YAN Z. Linear quadratic control of backward stochastic differential equation with partial information[J]. Applied Mathematics and Computation, 2021, 403: 126164. [38] SUN J, XIONG J. Stochastic linear-quadratic optimal control with partial observation[J]. arXiv preprint arXiv:2202.13632, 2022. [39] WU Z, ZHUANG Y. Linear-quadratic partially observed forward–backward stochastic differential games and its application in finance[J]. Applied Mathematics and Computation, 2018, 321: 577-592. [40] WANG G, WU Z, XIONG J. Maximum principles for forward-backward stochastic control systems with correlated state and observation noises[J]. SIAM Journal on Control and Optimization, 2013, 51(1): 491-524. [41] WANG G, XIAO H, XIONG J. A kind of lq non-zero sum differential game of backward stochastic differential equation with asymmetric information[J]. Automatica, 2018, 97: 346-352. [42] WANG G, XIAO H, XING G. An optimal control problem for mean-field forward–backward stochastic differential equation with noisy observation[J]. Automatica, 2017, 86: 104-109. [43] WONHAM W M. On the separation theorem of stochastic control[J]. SIAM Journal on Control, 1968, 6(2): 312-326. [44] BENSOUSSAN A. Estimation and control of dynamical systems: volume 48[M]. Springer, 2018. [45] KARATZAS I, SHREVE S. Brownian motion and stochastic calculus: volume 113[M]. Springer Science & Business Media, 2012. [46] ETHERIDGE A, BAXTER M. A course in financial calculus[M]. Cambridge University Press, 2002. [47] KARATZAS I, OCONE D L, LI J. An extension of clark’formula[J]. Stochastics: An International Journal of Probability and Stochastic Processes, 1991, 37(3): 127-131. [48] ROGERS L C G, WILLIAMS D. Diffusions, markov processes and martingales: Volume 2, itô calculus: volume 2[M]. Cambridge university press, 2000
所在学位评定分委会	数学系
国内图书分类号	O232
来源库	人工提交
成果类型	学位论文
条目标识符	http://sustech.caswiz.com/handle/2SGJ60CL/336417
专题	理学院_数学系
推荐引用方式 GB/T 7714	Ou YY. MEAN-FIELD LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS WITH PARTIAL OBSERVATION[D]. 深圳. 南方科技大学,2022.

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