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

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 ε, the parameter of the approximation, goes to zero. First, for a twofold junction problem we characterize the limit value function as a 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 a maximal subsolution.