# American Institute of Mathematical Sciences

September  2013, 8(3): 707-726. doi: 10.3934/nhm.2013.8.707

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

 1 ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

Received  April 2012 Revised  July 2013 Published  October 2013

A few applications of the viability theory to the solution to the Hamilton-Jacobi-Moskowitz problems are presented. In the considered problem 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 semi-explicit form for an homogeneous lagrangian do not hold, a capture basin algorithm is proposed 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 to traffic regulation problems.
Citation: Anya Désilles. Viability approach to Hamilton-Jacobi-Moskowitz problem involving variable regulation parameters. Networks & Heterogeneous Media, 2013, 8 (3) : 707-726. doi: 10.3934/nhm.2013.8.707
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