Biorthogonal functions for complex exponentials and an application to the controllability of the Kawahara equation via a moment approach

2022/11/30 by Ademir F. Pazoto, Pazoto, Ademir F., Miguel Soto +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2212.00158

openalex publication_date 2022/11/30 · openalex created_date 2022/12/13 · openalex updated_date 2026/07/28

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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