LaxHopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory

Christian G. Claudel, Alexandre M. Bayen

Research output: Contribution to journalArticlepeer-review

122 Scopus citations


This article proposes a new approach for computing a semi-explicit form of the solution to a class of HamiltonJacobi (HJ) partial differential equations (PDEs), using control techniques based on viability theory. We characterize the epigraph of the value function solving the HJ PDE as a capture basin of a target through an auxiliary dynamical system, called characteristic system. The properties of capture basins enable us to define components as building blocks of the solution to the HJ PDE in the Barron/Jensen-Frankowska sense. These components can encode initial conditions, boundary conditions, and internal boundary conditions, which are the topic of this article. A generalized Lax-Hopf formula is derived, and enables us to formulate the necessary and sufficient conditions for a mixed initial and boundary conditions problem with multiple internal boundary conditions to be well posed. We illustrate the capabilities of the method with a data assimilation problem for reconstruction of highway traffic flow using Lagrangian measurements generated from Next Generation Simulation (NGSIM) traffic data.

Original languageEnglish (US)
Article number5404403
Pages (from-to)1142-1157
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number5
StatePublished - May 2010
Externally publishedYes

Bibliographical note

Funding Information:
Manuscript received April 01, 2008; revised April 12, 2009. First published February 02, 2010; current version published May 12, 2010. This work was supported in part by technologies developed by the company VIMADES. Recommended by Associate Editor D. Dochain.


  • Hamilton-Jacobi (HJ)
  • Next generation simulation (NGSIM)
  • Partial differential equations (PDEs)

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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