An Efficient Numerical Algorithm for Solving Data Driven Feedback Control Problems
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An Efficient Numerical Algorithm for Solving Data Driven Feedback Control Problems Richard Archibald1 · Feng Bao2
· Jiongmin Yong3 · Tao Zhou4
Received: 1 August 2020 / Revised: 25 October 2020 / Accepted: 29 October 2020 / Published online: 10 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The goal of this paper is to solve a class of stochastic optimal control problems numerically, in which the state process is governed by an Itô type stochastic differential equation with control process entering both in the drift and the diffusion, and is observed partially. The optimal control of feedback form is determined based on the available observational data. We call this type of control problems the data driven feedback control. The computational framework that we introduce to solve such type of problems aims to find the best estimate for the optimal control as a conditional expectation given the observational information. To make our method feasible in providing timely feedback to the controlled system from data, we develop an efficient stochastic optimization algorithm to implement our computational framework. Keywords Stochastic optimal control · Nonlinear filtering · Data driven · Maximum principle · Stochastic optimization Mathematics Subject Classification 93E11 · 60G35 · 65K10
1 Introduction Stochastic optimal control is an important research subject that attracts scientists and engineers in various fields from theoretical scientific research to practical industrial production. The control process (also called control policy), which controls a stochastic dynamical system whose solution called state process, is designed to meet some optimality conditions. For the classic stochastic optimal control problem with full observation of the state, both theo-
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Feng Bao [email protected]
1
Computational Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA
2
Department of Mathematics, Florida State University, Tallahassee, FL, USA
3
Department of Mathematics, University of Central Florida, Orlando, FL, USA
4
Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, China
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Journal of Scientific Computing (2020) 85:51
retical results and numerical methods are extensively studied. However, in practice the full observation of the state process is often not available. Instead, we have detectors/observation facilities to collect partial observational data, which provide indirect information about the state process. The theoretical formulation for partially observable stochastic optimal control is derived analytically [14,20,26,30,40,42], and the corresponding optimal control is a stochastic process adapted to the observational information. Since the controlled state needs to be inferred from observations, the procedure of finding the optimal control requires data analysis for observational data, and the control actions are driven by information contained in data. To highlight the influence of
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