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Recent progress on controllability/observability for systems governed by partial differential equations

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Abstract

The main purpose of this paper is to overview some recent methods and results on controllability/observability problems for systems governed by partial differential equations. First, the authors review the theory for linear partial differential equations, including the iteration method for the null controllability of the time-invariant heat equation and the Rellich-type multiplier method for the exact controllability of the time-invariant wave equation, and especially a unified controllability/observability theory for parabolic and hyperbolic equations based on a global Carleman estimate. Then, the authors present sharp global controllability results for both semi-linear parabolic and hyperbolic equations, based on linearization approach, sharp observability estimates for the corresponding linearized systems and the fixed point argument. Finally, the authors survey the local null controllability result for a class of quasilinear parabolic equations based on the global Carleman estimate, and the local exact controllability result for general hyperbolic equations based on a new unbounded perturbation technique.

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Authors and Affiliations

  1. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Hongheng Li & Qi Lü

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

    Xu Zhang

Authors
  1. Hongheng Li
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  2. Qi Lü
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  3. Xu Zhang
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Corresponding author

Correspondence to Hongheng Li.

Additional information

This work is partially supported by the National Science Foundation of China under Grant Nos. 10831007, 60821091, and 60974035, and the project MTM2008-03541 of the Spanish Ministry of Science and Innovation.

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Li, H., Lü, Q. & Zhang, X. Recent progress on controllability/observability for systems governed by partial differential equations. J Syst Sci Complex 23, 527–545 (2010). https://doi.org/10.1007/s11424-010-0144-9

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  • DOI: https://doi.org/10.1007/s11424-010-0144-9

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