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Regularity of the Inertial Manifolds for Evolution Equations in Admissible Spaces and Finite-Dimensional Feedback Controllers

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Abstract

Our main aims in this paper are to investigate the regularity of inertial manifolds for non-autonomous semi-linear evolution equations and to give an application of inertial manifolds to a feedback control problem. We first prove that the inertial manifolds are smooth if the nonlinear term is smooth. Then, using the theory of inertial manifolds for non-autonomous semi-linear evolution equations, we construct a feedback controller for a class of control problems for the one-dimensional reaction-diffusion equations with the Lipschitz coefficient of the nonlinear term which may depend on time and belong to an admissible space.

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Acknowledgments

We would like to thank the referee for careful reading of our manuscript. His/her comments, remarks and corrections lead to the improvements of the paper. The second author would also like to express the deep gratitude to Viet Duoc Trinh for the fruitful discussions, and to Ricardo Rosa for the helpful discussions and for the useful document related to the paper R. Rosa and R. Temam [32].

Funding

The works of the first two authors are supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM).

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Authors and Affiliations

  1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vien Toan ung dung va Tin hoc, Dai hoc Bach Khoa Hanoi, 1 Dai Co Viet, Hanoi, Vietnam

    Thieu Huy Nguyen & Duc Thuan Do

  2. Faculty of Mathematics and Natural Sciences, Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam

    Xuan-Quang Bui

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  1. Thieu Huy Nguyen
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  2. Xuan-Quang Bui
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  3. Duc Thuan Do
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Correspondence to Thieu Huy Nguyen.

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Nguyen, T.H., Bui, XQ. & Do, D.T. Regularity of the Inertial Manifolds for Evolution Equations in Admissible Spaces and Finite-Dimensional Feedback Controllers. J Dyn Control Syst 28, 657–679 (2022). https://doi.org/10.1007/s10883-021-09538-1

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  • DOI: https://doi.org/10.1007/s10883-021-09538-1

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