Abstract
This article is concerned with the global exact controllability for ideal incompressible magnetohydrodynamics in a rectangular domain where the controls are situated in both vertical walls. First, global exact controllability via boundary controls is established for a related Elsässer type system by applying the return method, introduced in Coron [Math. Control Signals Syst. 5 (1992) 295–312]. Similar results are then inferred for the original magnetohydrodynamics system with the help of a special pressure-like corrector in the induction equation. Overall, the main difficulties stem from the nonlinear coupling between the fluid velocity and the magnetic field in combination with the aim of exactly controlling the system. In order to overcome some of the obstacles, we introduce ad-hoc constructions, such as suitable initial data extensions outside of the physical part of the domain and a certain weighted space.
Highlights
The property of global exact controllability by means of boundary controls has been proved for perfect fluids, which are described by the incompressible Euler equations, already more than two decades ago for various kinds of bounded domains under employment of the return method introduced by Coron in [7]
In the context of ideal incompressible magnetohydrodynamics, which contains the incompressible Euler system coupled in a nonlinear way with the induction equation for the evolution of the magnetic field, similar results on controllability are not available up to the present moment
The goal of this work is to make a first step in this direction for the very particular case of a rectangular domain
Summary
The property of global exact controllability by means of boundary controls has been proved for perfect fluids, which are described by the incompressible Euler equations, already more than two decades ago for various kinds of bounded domains under employment of the return method introduced by Coron in [7]. In the context of ideal incompressible magnetohydrodynamics, which contains the incompressible Euler system coupled in a nonlinear way with the induction equation for the evolution of the magnetic field, similar results on controllability are not available up to the present moment. In this article we shall prove the small-time global exact boundary controllability for the Elsasser system (1.3) whereby the obtained solutions may have the property ∇p+ = ∇p−. One possible approach for showing ∇q ≡ 0, which is under ongoing investigation, would be to establish suitable well-posedness results for (1.1) with non-characteristic boundary conditions at Γ0
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: ESAIM: Control, Optimisation and Calculus of Variations
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
