  We are interested in the feedback stabilization of systems described by
Hamilton-Jacobi type equations in $\mathbb{R}^n$. A reformulation leads to a a
stabilization problem for a multi-dimensional system of $n$ hyperbolic partial
differential equations. Using a novel Lyapunov function taking into account the
multi-dimensional geometry we show stabilization in $L^2$ for the arising
system using a suitable feedback control. We further present examples of such
systems partially based on a forming process.
