TY - JOUR
T1 - LaxHopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I
T2 - Theory
AU - Claudel, Christian G.
AU - Bayen, Alexandre M.
N1 - 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.
PY - 2010/5
Y1 - 2010/5
N2 - 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.
AB - 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.
KW - Hamilton-Jacobi (HJ)
KW - Next generation simulation (NGSIM)
KW - Partial differential equations (PDEs)
UR - http://www.scopus.com/inward/record.url?scp=77952196906&partnerID=8YFLogxK
U2 - 10.1109/TAC.2010.2041976
DO - 10.1109/TAC.2010.2041976
M3 - Article
AN - SCOPUS:77952196906
SN - 0018-9286
VL - 55
SP - 1142
EP - 1157
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
IS - 5
M1 - 5404403
ER -