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

TL;DR

This paper establishes the exact controllability of the Kawahara equation on a periodic domain using biorthogonal functions and a moment approach, with implications for control theory of nonlinear PDEs.

Contribution

It introduces a novel application of biorthogonal functions to control the Kawahara equation, extending controllability results to a nonlinear dispersive PDE.

Findings

Exact controllability of the linearized Kawahara system was proved.

Local controllability of the full nonlinear system was achieved.

A new method combining Fourier analysis and biorthogonal functions was developed.

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.