Abstract
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 language | English (US) |
---|---|
Article number | 5404403 |
Pages (from-to) | 1142-1157 |
Number of pages | 16 |
Journal | IEEE Transactions on Automatic Control |
Volume | 55 |
Issue number | 5 |
DOIs | |
State | Published - May 2010 |
Externally published | Yes |
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.
Keywords
- 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