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Observability and attractors of nonlinear Von Kármán beams

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Abstract

In this paper, observability inequalities and their influence on the dynamics of the solutions of nonlinear von Kármán beam are studied under nonlinear damping and external forces. The nonlinear von Kármán beam equations describe the nonlinear oscillations with large displacements. The nonlinear terms of the considered model pose new mathematical and technical difficulties in our analysis. Under quite general assumptions on nonlinear sources terms and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions to the considered problem. The main objective is to construct, from observability inequalities, a smooth global attractor and a generalized exponential attractor with finite fractal dimensions. This paper is the first to study the dynamics of the von Kármán beam from observability inequalities. The results obtained generalize and improve some previous results and can also be used in control theory.

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Acknowledgements

The authors would like to thank the Editor of NoDEA Prof. Alessandra Lunardi and the anonymous reviewer for their recommendations and remarks aiming at improving the manuscript in terms of clarity.

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Authors and Affiliations

  1. Université de Carthage, Ecole Nationale d’Ingénieurs de Bizerte, BP66, 7035, Bizerte, Tunisia

    Moncef Aouadi

  2. UR Systèmes dynamiques et Applications, UR, 17ES21, Bizerte, Tunisia

    Moncef Aouadi & Souad Guerine

  3. Faculté des Sciences de Bizerte, Université de Carthage, 7021, Jarzouna, Tunisia

    Souad Guerine

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  1. Moncef Aouadi
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  2. Souad Guerine
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Correspondence to Moncef Aouadi.

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Aouadi, M., Guerine, S. Observability and attractors of nonlinear Von Kármán beams. Nonlinear Differ. Equ. Appl. 30, 70 (2023). https://doi.org/10.1007/s00030-023-00873-9

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