Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker-Planck equation - Sorbonne Université Access content directly
Preprints, Working Papers, ... Year : 2023

Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker-Planck equation

Abstract

We study the exact controllability of the evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0 \end{equation*} where $A$ is a nonnegative self-adjoint operator on a Hilbert space $X$ and $B$ is an unbounded linear operator on $X$, which is dominated by the square root of $A$. The control action is bilinear and only of scalar-input form, meaning that the control is the scalar function $p$, which is assumed to depend only on time. Furthermore, we only consider square-integrable controls. Our main result is the local exact controllability of the above equation to the ground state solution, that is, the evolution through time, of the first eigenfunction of $A$, as initial data. The analogous problem (in a more general form) was addressed in our previous paper [Exact controllablity to eigensolutions for evolution equations of parabolic type via bilinear control, Alabau-Boussouira F., Cannarsa P. and Urbani C., Nonlinear Diff. Eq. Appl. (2022)] for a bounded operator $B$. The current extension to unbounded operators allows for many more applications, including the Fokker-Planck equation in one space dimension, and a larger class of control actions.

Dates and versions

hal-04021363 , version 1 (09-03-2023)

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Fatiha Alabau-Boussouira, Piermarco Cannarsa, Cristina Urbani. Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker-Planck equation. 2023. ⟨hal-04021363⟩
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