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

TL;DR

This paper investigates optimal control problems on networks with junctions, using thermostatic approximations to analyze the limit behavior of the value function as the approximation parameter approaches zero.

Contribution

It introduces a novel approximation method for junctions in network control problems and characterizes the limit solutions as viscosity solutions, providing new uniqueness results.

Findings

Limit value functions characterized as viscosity solutions.

Established maximal subsolution properties for junction problems.

Proved uniqueness of solutions in the approximated junction models.

Abstract

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.