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

Hybrid Thermostatic Approximations of Junctions for some Optimal Control Problems on Networks