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

Fabio Bagagiolo, Rosario Maggistro

Published 2017-10-19Version 1

The aim of this paper is to study some optimal control problems on networks with junctions. We consider an approximation of such a problems given by the use of 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, we take into account a twofold junction problem for which we characterize the limit function as viscosity solution and maximal subsolution of a suitable Hamilton-Jacobi problem. Afterwards we analyse a threefold junction problem for which we consider two different way for passing to the limit in its approximation. For both ways we recover some uniqueness results in the sense of maximal subsolution.