Optimal control for uncertain discrete-time singular systems under expected value criterion
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Optimal control for uncertain discrete-time singular systems under expected value criterion Yadong Shu1
· Bo Li2 · Yuanguo Zhu3
Accepted: 24 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Optimal control problems governed by two different types of uncertain discrete-time singular systems are investigated under expected value criterion. The objective function including uncertain variables is optimized with the help of expected value method provided that the singular systems are both regular and impulse-free. At first, based on the principle of dynamic programming, a recurrence equation is derived to simplify an optimal control model for a class of uncertain discrete-time singular systems. After that, according to uncertainty theory and the recurrence equation, two kinds of optimal control problems subject to an uncertain linear singular system and an uncertain singular system with quadratic input variables are considered in order, and the optimal solutions are both presented by accurate expressions. A numerical example and a dynamic input-output model are settled to illustrate the effectiveness of the results obtained. Keywords Optimal control · Uncertain singular systems · Expected value · Recurrence equation
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Yadong Shu [email protected]; [email protected] Bo Li [email protected] Yuanguo Zhu [email protected]
1
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2
School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China
3
School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
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Y. Shu et al.
1 Introduction In the middle of 20th century, the investigations on optimal control problems began to attract the attention of so many researchers for the necessity of strict expression form in optimal control theory. With the rapid development in the fields of computer science and modern mathematics, optimal control theory has achieved lots of meaningful progresses at the theoretical level (Kirk 2012), and also in practical areas (Konstantin et al. 2015) such as economics, management, space technology and production engineering. An optimal control problem ruled by a discrete-time singular system is to select the best input so that the given objective function, also called performance index, takes the optimal value. Such type of problems also have plenty of applications in various domains. From 1960s, stochastic optimal control started to be discussed by some mathematicians, such as Merton (1971) and Harrison (1985). Then, several valuable investigations on optimal control of stochastic differential equations and applications of stochastic systems for finance can be seen in many published articles. Furthermore, optimal control models governed by other kinds of stochastic systems including switched stochastic systems and time-delay stochastic systems have been widely researched. Zhang et al. (2010) considered an optimal control
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