Exact controllability of anisotropic 1D partial differential equations in spaces of analytic functions

TL;DR

This paper establishes local exact controllability for a broad class of 1D PDEs in spaces of analytic functions, using a nonlinear Cauchy problem approach and analyzing derivatives' jets.

Contribution

It introduces a novel controllability result for 1D PDEs with analytic initial and target states, including nonlinear and boundary conditions.

Findings

Controllability proven for various PDEs like Boussinesq, Ginzburg-Landau, Kuramoto-Sivashinsky, KdV.

Uses Gevrey spaces and jet analysis to establish controllability.

Results applicable to PDEs with boundary conditions and nonlinear terms.

Abstract

In this article, we prove a local controllability result for a general class of 1D partial differential equations on the interval $(0,1)$. The PDEs we consider take the form $\partial_t^N y=\zeta_M \partial_{x}^{M}y+f(x , y , \partial_{x} y,\ldots, \partial_x^{M-1} y)$ where $1\le N < M$, $\zeta_M\in \mathbb{C} ^*$, and $f$ is some linear or nonlinear term of lower order. In this context, we prove a local controllability result between states that are analytic functions. If some boundary conditions are prescribed, a similar local controllability result holds between analytic functions satisfying some compatibility conditions that are natural for the existence of smooth solutions of the considered PDE. The proof is performed by studying a nonlinear Cauchy problem in the spatial variable with data in some spaces of Gevrey functions and by investigating the relationship between the jet of space derivatives and the jet of time derivatives. We give various examples of applications, including the (good and bad) Boussinesq equation, the Ginzburg-Landau equation, the Kuramoto-Sivashinsky equation and the Korteweg-de Vries equation.