We are interested in the feedback stabilization of a quantity which evolution can be described using a Hamilton-Jacobi equation in $R^n$. The mathematical treatment then leads to a system of $n$ hyperbolic transport PDEs for the perturbation of a desired state which should be stabilized. There exists a rich literature for the one dimensional case including different Lyapunov functions leading to exponential decay of the $L^2$ norm when suitable feedback controls are applied. Here we want to extend these results to the multi-dimensional case leading to a novel Lyapunov function with space dependent weight functions taking accounting for the multidimensional geometry. We show the exponential decay of the Lyapunov function provided that the weights and controls are chosen appropriately. The design feedback control is closely related to the precise choice of the Lyapunov function and its weights. We further present numerical experiments.