  We show a large time behavior result for class of weakly coupled systems of
first-order Hamilton-Jacobi equations in the periodic setting. We use a PDE
approach to extend the convergence result proved by Namah and Roquejoffre
(1999) in the scalar case. Our proof is based on new comparison, existence and
regularity results for systems. An interpretation of the solution of the system
in terms of an optimal control problem with switching is given.
