Pré-Publication, Document De Travail Année : 2025

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

Résumé

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 $\partial_t^N y=\zeta_M \partial_{x}^{M}y+f(x , y , \partial_{x} y,..., \partial _x^{M-1} y)$ 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.
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Dates et versions

hal-04931607 , version 1 (05-02-2025)

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Camille Laurent, Ivonne Rivas, Lionel Rosier. Exact controllability of anisotropic 1D partial differential equations in spaces of analytic functions. 2025. ⟨hal-04931607⟩
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