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Exponential Dichotomy and Stable Manifolds for Differential-Algebraic Equations on the Half-Line

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Abstract

We study linear and semi-linear differential-algebraic equations (DAEs) on the half-line \(\mathbb {R}_{+}\). In our strategy, we first show a characterization for the existence of exponential dichotomy for linear DAEs based on the Lyapunov-Perron method. Then, we prove the existence of invariant stable manifolds for semi-linear DAEs in the case that the evolution family corresponding to a linear DAE admits an exponential dichotomy and the non-linear forcing function fulfills the non-uniform φ-Lipschitz condition where the Lipschitz function φ belongs to wide classes of admissible function spaces such as Lp, \(1\leq p \leq \infty \), and Lp,q.

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Acknowledgements

We thank the reviewers for their careful reading and insightful assessment of our work.

Funding

The first author is financially supported by the Vietnam National University (Hanoi) under the research project QG 21.03. The second author is financially supported by the Vietnam National Foundation for Science and Technology Development under Grant No. 101.02-2021.04.

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Authors and Affiliations

  1. Faculty of Mathematics-Mechanics-Informatics, Hanoi University of Science, Khoa Toan-Co-Tin hoc, Dai Hoc Khoa Hoc Tu Nhien, DHQGHN, 334 Nguyen Trai St., Hanoi, Vietnam

    Phi Ha

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

    Thieu Huy Nguyen

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  1. Phi Ha
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Correspondence to Thieu Huy Nguyen.

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Ha, P., Nguyen, T.H. Exponential Dichotomy and Stable Manifolds for Differential-Algebraic Equations on the Half-Line. J Dyn Control Syst 29, 475–500 (2023). https://doi.org/10.1007/s10883-022-09596-z

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  • DOI: https://doi.org/10.1007/s10883-022-09596-z

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