Viability approach to Hamilton-Jacobi-Moskowitz problem involving variable regulation parameters

Abstract

We present a few applications of the viability theory to the solution to the Hamilton-Jacobi-Moskowitz problems when the Hamiltonian (fundamental diagram) depends on time, position and/or some regulation parameters. We study such a problem in its equivalent variational formulation. In this case, the corresponding lagrangian depends on the state of the characteristic dynamical system. As the Lax-Hopf formulae that give the solution in a semiexplicit form for an homogeneous lagrangian do not hold, we use a capture basin algorithm to compute the Moskowitz function as a viability solution of the Hamilton-Jacobi-Moskowitz problem with general conditions (including initial, boundary and internal conditions). We present two examples of applications.In the first one we introduce the variable speed limit as a regulation parameter. Our approach allows to compute the Moscowitz function for all values of the variable speed limit in a selected range and then to analyze its influence on the traffic flow. In particular, we study the case when the variation of the speed limit is applied locally, in space and time.Our second example deals with the local load capacity variations on the road. Such a variation can be a permanent property of the road (road narrowing) or it can be due to a temporary change of number of lanes (an accident or roadworks, for example). One can also use it as a regulation parameter by variable assignment of a supplementary lane.

Transports Regulators of Networks with Junctions Detected by Durations Functions

“…The search of celerities is a by-product of most studies of road traffic. Greenshields used in 1933 photographic measurement methods for the first time to describe a phenomenological law described by a quadratic relation between vehicles and their density and flows, and using it as inputs of first-order partial differential equations (conservation laws and Hamilton-Jacobi-Bellman equations) providing "cumulative number" of vehicles passing at given position after a given time on a one-dimensional road (see [ [26,27,Claudel,Bayen]), [33,Désilles].…”