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Abstract
This article studies stabilisation problem for time-varying linear systems via state feedback. Two types of controllers are designed by utilising solutions to Lyapunov differential equations. The first type of feedback controllers involves the unique positive-definite solution to a parametric Lyapunov differential equation, which can be solved when either the state transition matrix of the open-loop system is exactly known, or the future information of the system matrices are accessible in advance. Different from the first class of controllers which may be difficult to implement in practice, the second type of controllers can be easily implemented by solving a state-dependent Lyapunov differential equation with a given positive-definite initial condition. In both cases, explicit conditions are obtained to guarantee the exponentially asymptotic stability of the associated closed-loop systems. Numerical examples show the effectiveness of the proposed approaches.
Acknowledgements
The authors thank the Associate Editor and the anonymous reviewers for their helpful comments and suggestions which have helped to improve the quality of this article. This work was supported in part by the National Natural Science Foundation of China under grant numbers 61273028, 61203007 and 60904007, by the Fundamental Research Funds for the Central Universities under Grant HIT.BRETIII.201210, by the Program for New Century Excellent Talents in University under Grant NCET-11-0815, and by the Foundation for Innovative Research Group of the National Natural Science Foundation of China under Grant 61021002.