Sequential adaptive switching time optimization technique for maximum hands-off control problems

Sida Lin; Lixia Meng; Jinlong Yuan; Changzhi Wu; An Li; Chongyang Liu; Jun Xie; Sida Lin; Lixia Meng; Jinlong Yuan; Changzhi Wu; An Li; Chongyang Liu; Jun Xie

doi:10.3934/era.2024101

Electronic Research Archive

2024, Volume 32, Issue 4: 2229-2250. doi: 10.3934/era.2024101

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Sequential adaptive switching time optimization technique for maximum hands-off control problems

1.
Department of Applied Mathematics, School of Science, Dalian Maritime University, Dalian 116026, China
2.
Chongqing National Center for Applied Mathematics, Chongqing Normal University, Chongqing 404087, China
3.
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
4.
School of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, China
5.
Yantai Key Laboratory of Big Data Modeling and Intelligent Computing, Yantai 264005, China
6.
Department of Basics, PLA Dalian Naval Academy, Dalian 116018, China

Received: 11 January 2024 Revised: 01 March 2024 Accepted: 13 March 2024 Published: 19 March 2024

In this paper, we consider maximum hands-off control problem governed by a nonlinear dynamical system, where the maximum hands-off control constraint is characterized by an $ L^{0} $ norm. For this problem, we first approximate the $ L^{0} $ norm constraint by a $ L^{1} $ norm constraint. Then, the control parameterization together with sequential adaptive switching time optimization technique is proposed to approximate the optimal control problem by a sequence of finite-dimensional optimization problems. Furthermore, a smoothing technique is exploited to approximate the non-smooth maximum operator and an error analysis is investigated for this approximation. The gradients of the cost functional with respect to the decision variables in the approximate problem are derived. On the basis of these results, we develop a gradient-based optimization algorithm to solve the resulting optimization problem. Finally, an example is solved to demonstrate the effectiveness of the proposed algorithm.
- maximum hands-off control,
- control parameterization,
- sequential adaptive switching time optimization technique,
- smoothing technique,
- error analysis
Citation: Sida Lin, Lixia Meng, Jinlong Yuan, Changzhi Wu, An Li, Chongyang Liu, Jun Xie. Sequential adaptive switching time optimization technique for maximum hands-off control problems[J]. Electronic Research Archive, 2024, 32(4): 2229-2250. doi: 10.3934/era.2024101

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Abstract

In this paper, we consider maximum hands-off control problem governed by a nonlinear dynamical system, where the maximum hands-off control constraint is characterized by an $ L^{0} $ norm. For this problem, we first approximate the $ L^{0} $ norm constraint by a $ L^{1} $ norm constraint. Then, the control parameterization together with sequential adaptive switching time optimization technique is proposed to approximate the optimal control problem by a sequence of finite-dimensional optimization problems. Furthermore, a smoothing technique is exploited to approximate the non-smooth maximum operator and an error analysis is investigated for this approximation. The gradients of the cost functional with respect to the decision variables in the approximate problem are derived. On the basis of these results, we develop a gradient-based optimization algorithm to solve the resulting optimization problem. Finally, an example is solved to demonstrate the effectiveness of the proposed algorithm.

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