We consider the three-dimensional ideal MHD system on a domain \(\Omega ' \subset \mathbb {R}^3\) with an open subset \(\Gamma \) of the boundary \(\partial \Omega '\) where we prescribe both \(u\cdot \textsf{n}\) and \(b\cdot \textsf{n}\) , while \(u\cdot \textsf{n}= b\cdot \textsf{n}=0\) on \(\partial \Omega ' \setminus \Gamma \) . We prove boundary controllability of the system, namely that we can prescribe the boundary data such that the unique solution of the system with an initial state \((u_0,b_0)\) achieves another state \((u_1,b_1)\) in a finite time, where \(u_0,b_0,u_1,b_1\) are arbitrary divergence-free vector fields satisfying impermeability boundary condition and which are extendible to vector fields with the same properties on any bounded domain obtained by extension of \(\Omega '\) via \(\Gamma \) . As a byproduct, we give the first local well-posedness proof for the incompressible ideal MHD system which does not use Elsasser variables and is thus applicable to any bounded domain with sufficient Sobolev regularity. We also provide a new, simpler control for the two-dimensional ideal MHD system.
