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

Camille Laurent, Ivonne Rivas, Lionel Rosier

TL;DR

This work addresses exact controllability for a broad class of anisotropic 1D PDEs on $[0,1]$ with higher spatial order than temporal order, proving local controllability between analytic states under natural compatibility conditions. The authors develop a nonlinear Cauchy problem in the spatial variable within Gevrey spaces and establish a precise jet-derivative correspondence between space and time to enable forward and backward evolution. They prove a local controllability result with explicit Gevrey regularity $y\in G^{1,\lambda}([0,1]\times[0,T])$ (where $\lambda=M/N>1$) and provide a comprehensive set of applications to models such as KdV, Boussinesq, complex Ginzburg-Landau, and Kuramoto-Sivashinsky, as well as linear constant-coefficient PDEs. The paper also develops a Gevrey-functional framework, compatibility sets, and a complex-analytic toolkit, enabling the extension of the approach to complex-valued equations. Overall, the work advances controllability theory for ill-posed-looking problems by exploiting spatial Cauchy-analysis and analytic data to achieve precise, high-regularity controllability results.

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 $\partial_t^N y=ζ_M \partial_{x}^{M}y+f(x , y , \partial_{x} y,\ldots, \partial_x^{M-1} y)$ where $1\le N < M$, $ζ_M\in \mathbb{C} ^*$, 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.

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

TL;DR

This work addresses exact controllability for a broad class of anisotropic 1D PDEs on with higher spatial order than temporal order, proving local controllability between analytic states under natural compatibility conditions. The authors develop a nonlinear Cauchy problem in the spatial variable within Gevrey spaces and establish a precise jet-derivative correspondence between space and time to enable forward and backward evolution. They prove a local controllability result with explicit Gevrey regularity (where ) and provide a comprehensive set of applications to models such as KdV, Boussinesq, complex Ginzburg-Landau, and Kuramoto-Sivashinsky, as well as linear constant-coefficient PDEs. The paper also develops a Gevrey-functional framework, compatibility sets, and a complex-analytic toolkit, enabling the extension of the approach to complex-valued equations. Overall, the work advances controllability theory for ill-posed-looking problems by exploiting spatial Cauchy-analysis and analytic data to achieve precise, high-regularity controllability results.

Abstract

In this article, we prove a local controllability result for a general class of 1D partial differential equations on the interval . The PDEs we consider take the form where , , and 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.

Paper Structure

This paper contains 19 sections, 39 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. The Korteweg-de Vries equation
  4. Boussinesq equation
  5. Neumann boundary conditions
  6. Dirichlet boundary conditions
  7. The complex Ginzburg-Landau equation
  8. The Kuramoto-Sivashinsky equation
  9. The case of a linear PDE with constant coefficients
  10. Cauchy problem in the space variable
  11. Statement of the global wellposedness result
  12. Abstract existence theorem
  13. Gevrey type functional spaces
  14. Application to the semi-linear PDE
  15. Correspondence between the space derivatives and the time derivatives
  16. ...and 4 more sections

Key Result

Lemma 1.1

For any $l\in \mathbb N$, there exist a number $m=m(l)\in \mathbb N$ and a smooth application $J_{l}: [-1,1]\times (\mathbb R^{N})^{m(l)+1}\to \mathbb R^{M}$ such that for any solution $y\in C^\infty ([-1,1]\times [t_1,t_2])$ of $\partial _t^N y = P\, y + f(x , y , \partial_{x} y,..., \partial _x^{

Theorems & Definitions (78)

  • Lemma 1.1
  • Definition 1
  • Theorem 1.2
  • Proposition 1.3
  • Remark 1
  • Lemma 2.1
  • proof
  • Remark 2
  • Theorem 2.2
  • Theorem 2.3
  • ...and 68 more