Exact boundary controllability of the 3D incompressible ideal MHD system

We consider the three-dimensional ideal MHD system on a domain ${\mathrm{\Omega }}^{\mathrm{\prime }}\subset {\mathbb{R}}^3$ with a part $\mathrm{\Gamma }$ of the boundary~$\mathrm{\partial }\mathrm{\Omega }$, where we prescribe both $u\cdot n$ and $b\cdot n$, while $u\cdot n=b\cdot n=0$ on $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}\setminus \mathrm{\Gamma }$. We prove the boundary controllability of the system, namely that we can prescribe the boundary data such that the unique solution of the system with initial state $(u_0,b_0)$ achieves another state $(u_1,b_1)$ in finite time, where $u_0,b_0,u_1,b_1$ are arbitrary divergence-free vector fields satisfying impermeability boundary condition which are extendable to vector fields with the same properties on any bounded domain obtained by extension of ${\mathrm{\Omega }}^{\mathrm{\prime }}$ via $\mathrm{\Gamma }$. As a byproduct, we give the first local well-posedness proof of 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 and simple proof of the $2$D controllability.