This talk describes the construction of lower semi continuous solutions to a specific class of Hamilton–Jacobi equations with concave Hamiltonians. The problem is posed as a capture basin problem, in which the target is defined in epigraphical form to encode the boundary and initial conditions of the problem. The solution is computed through its epigraph, which is the capture basin of the target under a set valued dynamics involving the Fenchel transform of the Hamiltonian. The resulting solution is a Barron–Jensen / Frankowska solution, which is lower semi continuous. The target is generalized to functions which enable one to include conditions inside the domain (internal boundary conditions), which in the context of traffic engineering can be interpreted as vehicle trajectories. A Lax–Hopf formula enables the computation of the solutions, which can be carried out explicitly in the case of piecewise affine conditions.
The resulting algorithms are applied to the problem of Lagrangian data assimilation for hydrodynamic models of traffic. Results from experimental deployments in California and New York will be also presented, as well as preliminary results from the Mobile Millennium field operational test in California (http://traffic.berkeley.edu), which is planned to reach 10,000 probe vehicles in a few months.