Global stabilisation of damped-driven conservation laws by a one-dimensional forcing

We study a multidimensional conservation law in a bounded domain, subject to a damping and an external force. Imposing the Dirichlet boundary condition and using standard methods of parabolic PDEs, it is straightforward to check that all the solutions are bounded in a Hölder space. Our main result proves that any trajectory can be exponentially stabilised by a one-dimensional external force supported in a given open subset. As a consequence, we obtain the  global approximate controllability to trajectories by a one-dimensional localised control. The proofs are based on the strong dissipation property of the PDEs in question and the theory of positivity preserving semigroups.