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

Anya Désilles

doi:10.3934/nhm.2013.8.707

Article Contents

2013, Volume 8, Issue 3: 707-726. Doi: 10.3934/nhm.2013.8.707

This issue Previous Article Numerical discretization of Hamilton--Jacobi equations on networks Next Article Explicit construction of solutions to the Burgers equation with discontinuous initial-boundary conditions

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

Anya Désilles^1,

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

Received: April 2012

Revised: July 2013

Published: September 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 49J40, 34A60; Secondary: 35L65.

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