  The control of relaxation-type systems of ordinary differential equations is
investigated using the Hamilton-Jacobi-Bellman equation. First, we recast the
model as a singularly perturbed dynamics which we embed in a family of
controlled systems. Then we study this dynamics together with the value
function of the associated optimal control problem. We provide an asymptotic
expansion in the relaxation parameter of the value function. We also show that
its solution converges toward the solution of a Hamilton-Jacobi-Bellman
equation for a reduced control problem. Such systems are motivated by
semi-discretisation of kinetic and hyperbolic partial differential equations.
Several examples are presented including Jin-Xin relaxation.
