  We study some optimal control problems on networks with junctions,
approximate the junctions by a switching rule of delay-relay type and study the
passage to the limit when $\varepsilon$, the parameter of the approximation,
goes to zero. First, for a twofold junction problem we characterize the limit
value function as viscosity solution and maximal subsolution of a suitable
Hamilton-Jacobi problem. Then, for a threefold junction problem we consider two
different approximations, recovering in both cases some uniqueness results in
the sense of maximal subsolution.
