Weak Closed-loop Solvability for Discrete-time Linear-Quadratic Optimal Control
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| Format: | Preprint |
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2025
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| _version_ | 1866916617766567936 |
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| author | Sun, Yue Wu, Xianping Li, Xun |
| author_facet | Sun, Yue Wu, Xianping Li, Xun |
| contents | In this paper, the open-loop, closed-loop, and weak closed-loop solvability for discrete-time linear-quadratic (LQ) control problem is considered due to the fact that it is always open-loop optimal solvable if the LQ control problem is closed-loop optimal solvable but not vice versa. The contributions are two-fold. On the one hand, the equivalent relationship between the closed-loop optimal solvability and the solution of the generalized Riccati equation is given. On the other hand, when the system is merely open-loop solvable, we have found the equivalent existence form of the optimal solution by perturbation method, which is said to be a weak closed-loop solution. Moreover, it obtains that there is an open-loop optimal control with a linear feedback form of the state. The essential technique is to solve the forward and backward difference equations by iteration. An example sheds light on the theoretical results established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11415 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak Closed-loop Solvability for Discrete-time Linear-Quadratic Optimal Control Sun, Yue Wu, Xianping Li, Xun Optimization and Control org In this paper, the open-loop, closed-loop, and weak closed-loop solvability for discrete-time linear-quadratic (LQ) control problem is considered due to the fact that it is always open-loop optimal solvable if the LQ control problem is closed-loop optimal solvable but not vice versa. The contributions are two-fold. On the one hand, the equivalent relationship between the closed-loop optimal solvability and the solution of the generalized Riccati equation is given. On the other hand, when the system is merely open-loop solvable, we have found the equivalent existence form of the optimal solution by perturbation method, which is said to be a weak closed-loop solution. Moreover, it obtains that there is an open-loop optimal control with a linear feedback form of the state. The essential technique is to solve the forward and backward difference equations by iteration. An example sheds light on the theoretical results established. |
| title | Weak Closed-loop Solvability for Discrete-time Linear-Quadratic Optimal Control |
| topic | Optimization and Control org |
| url | https://arxiv.org/abs/2502.11415 |