### Brockett’s First Example: An FAS Approach Treatment

DUAN Guang-Ren

1. Center for Control Science and Technology, Southern University of Science and Technology, Shenzhen 518055, China;Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China
• Received:2022-02-13 Published:2022-04-13
• Supported by:
This paper has been partially supported by the Major Program of National Natural Science Foundation of China under Grant Nos. 61690210, 61690212, the National Natural Science Foundation of China under Grant No. 61333003, and also by the Science Center Program of the National Natural Science Foundation of China under Grant No. 62188101.

DUAN Guang-Ren. Brockett’s First Example: An FAS Approach Treatment[J]. Journal of Systems Science and Complexity, 2022, 35(2): 441-456.

In this note, the well-known Brockett’s first example system is treated with the fully actuated system (FAS) approach. Firstly, it is shown that the system can be exponentially substabilized by a smooth controller in the sense that, except those starting from initial values on the z0-axis of the initial value space, all trajectories of the designed system as well as the control signals decay to zero exponentially. Secondly, global stabilization is realized through a way of enabling the trajectories starting from initial values on the z0-axis also to go to the origin. The idea is to firstly move an initial point on the z0-axis away from the axis using a pre-controller, and then to take over by the designed exponentially sub-stabilizing controller.
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