Speaker
Description
I will describe my recent contribution to the theory of Hamilton-Jacobi equations and optimal control problems posed on networks (see [1]).
I will study evolutive Hamilton–Jacobi equations with Hamiltonians that are discontinuous in time, posed on a simple network consisting of two edges on the real line connected at a single junction. I will introduce a notion of (flux-limited) viscosity solution for Hamiltonians H_i=H_i(t,x,p), (i=1,2) that are assumed to be only measurable in t. Moreover the flux limiter, A=A(t), acting at the junction, is not required to be continuous but only in L-infinity.
My first motivation is the optimal control problem studied by P. Cardaliaguet et P. Souganidis in [2], where they investigate how to optimally control traffic flow through a junction by acting only on the junction (i.e. with speed reduction or a traffic light) using a merely measurable flux limiter A(t).
In the case of convex Hamiltonians, I prove a comparison principle and establish an existence result via the construction of an optimal control problem. Generalizations to the nonconvex case and to more general networks are also discussed.
[1] A. Briani, On Hamilton Jacobi equations with time measurable Hamiltonians posed on a 1-dimensional junction , Preprint (2026), ArXiv https://arxiv.org/abs/2603.04183.
[2] P. Cardaliaguet, P. E. Souganidis, An optimal control problem of traffic flow on a junction, preprint, (2023), https://arxiv.org/abs/2312.15418.