Abstract
The paper deals with the controllability properties of the Kawahara equation posed on a periodic domain. We show that the equation is exactly controllable by means of a control depending only on time and acting on the system through a given shape function in space. Firstly, the exact controllability property is established for the linearized system through a Fourier expansion of solutions and the analysis of a biorthogonal sequence to a family of complex exponential functions. Finally, the local controllability of the full system is derived by combining the analysis of the linearized system, a fixed point argument and some Bourgain smoothing properties of the Kawahara equation on a periodic domain.
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Acknowledgements
We thank the anonymous referees for their valuable comments and remarks which have been taken into account in the revised version of the manuscript. The first author was partially support by CNPq (Brazil). The second author was supported by CAPES and CNPq (Brazil) and COLCIENCIAS, MINCIENCIAS885/2020 (Colombia).
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Pazoto, A.F., Vieira, M.D.S. Biorthogonal Functions for Complex Exponentials and an Application to the Controllability of the Kawahara Equation Via a Moment Approach. Appl Math Optim 88, 57 (2023). https://doi.org/10.1007/s00245-023-10032-2
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DOI: https://doi.org/10.1007/s00245-023-10032-2