On the global optimal solution for linear quadratic problems of switched system

Jin Feng He; Wei Xu; Zhi Guo Feng; Xinsong Yang

doi:10.3934/jimo.2018072

Article Contents

2019, Volume 15, Issue 2: 817-832. Doi: 10.3934/jimo.2018072

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On the global optimal solution for linear quadratic problems of switched system

1.
College of Mathematics Science, Chongqing Normal University, Chongqing, China
2.
School of Management, Shanghai University, Shanghai, China
3.
Faculty of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, China

* Corresponding author: Zhi Guo Feng
* Corresponding author: Zhi Guo Feng

Received: December 2016

Revised: March 2018

Early access: June 2018

Published: January 2019

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Abstract

The global optimal solution for the optimal switching problem is considered in discrete time, where these subsystems are linear and the cost functional is quadratic. The optimal switching problem is a discrete optimization problem. Complete enumeration search is always required to find the global optimal solution, which is very expensive. Relaxation method is an effective method to transform the discrete optimization problem into the continuous optimization problem, while the optimal solution is always not the feasible solution of the discrete optimization problem. In this paper, we propose a special class of relaxation method to transform the optimal switching problem into a relaxed optimization problem. We prove that the optimal solution of this modified relaxed optimization problem is exactly that of the optimal switching problem. Then, the global optimal solution can be obtained by solving the continuous optimization problem easily. Numerical examples are demonstrated to show that the modified relaxation method is efficient and effective to obtain the global optimal solution.

Keywords:

Mathematics Subject Classification: Primary: 49M20, 34K35.

Citation:

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Figure 1. Optimal weight of the first subsystem obtained by relaxation method and modified relaxation method in Example 1

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Figure 2. Optimal weight of the second subsystem obtained by relaxation method and modified relaxation method in Example 1

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Figure 3. Optimal weight of the third subsystem obtained by relaxation method and modified relaxation method in Example 1

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Figure 4. Optimal solution and truncated solution in Example 1

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Figure 5. Optimal weight of the first subsystem obtained by relaxation method and modified relaxation method in Example 2

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Figure 6. Optimal weight of the second subsystem obtained by relaxation method and modified relaxation method in Example 2

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Figure 7. Optimal weight of the third subsystem obtained by relaxation method and modified relaxation method in Example 2

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Figure 8. Optimal solution and truncated solution in Example 2

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