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Lyapunov stability and generalized invariance principle for nonconvex differential inclusions

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Abstract

This paper studies the system stability problems of a class of nonconvex differential inclusions. At first, a basic stability result is obtained by virtue of locally Lipschitz continuous Lyapunov functions. Moreover, a generalized invariance principle and related attraction conditions are proposed and proved to overcome the technical difficulties due to the absence of convexity. In the technical analysis, a novel set-valued derivative is proposed to deal with nonsmooth systems and nonsmooth Lyapunov functions. Additionally, the obtained results are consistent with the existing ones in the case of convex differential inclusions with regular Lyapunov functions. Finally, illustrative examples are given to show the effectiveness of the methods.

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Authors and Affiliations

  1. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Shu Liang, Xianlin Zeng & Yiguang Hong

Authors
  1. Shu Liang
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  2. Xianlin Zeng
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  3. Yiguang Hong
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Corresponding author

Correspondence to Yiguang Hong.

Additional information

This work was supported by the Beijing Natural Science Foundation (No. 4152057), the Natural Science Foundation of China (Nos. 61333001, 61573344), and the China Postdoctoral Science Foundation (No. 2015M581190).

Shu LIANG received the B.E. degree in Automatic Control and the Ph.D. degree in Engineering from the University of Science and Technology of China, Hefei, China, in 2010 and 2015, respectively. He is currently a postdoctoral fellow with the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China. His research interests include nonsmooth systems and control, distributed optimizations, game theory, and fractional order systems.

Xianlin ZENG received the B.S. and M.S. degrees in Control Science and Engineering from the Harbin Institute of Technology, Harbin, China, in 2009 and 2011, respectively, and the Ph.D. degree in Mechanical Engineering from the Texas Tech University. He is currently a postdoctoral in Key Laboratory of Systems and Control, Institute of Systems Science, Chinese Academy of Sciences, Beijing, China. His current research interests include distributed optimization and distributed control, hybrid control and synchronization of network systems.

Yiguang HONG received his B.S. and M.S. degrees from Department of Mechanics, Peking University, China, and Ph.D. degree from Chinese Academy of Sciences (CAS). He is currently a professor in Academy of Mathematics and Systems Science, CAS. His research interests include nonlinear dynamics and control, multi-agent systems, distributed optimization, and social networks.

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Liang, S., Zeng, X. & Hong, Y. Lyapunov stability and generalized invariance principle for nonconvex differential inclusions. Control Theory Technol. 14, 140–150 (2016). https://doi.org/10.1007/s11768-016-6037-2

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  • DOI: https://doi.org/10.1007/s11768-016-6037-2

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