Data

A Time-Dependent Hamilton-Jacobi Formulation of Reachable Sets for Continuous Dynamic Games

Mitchell, I.M.
Bayen, A.M.
Tomlin, C.J.
2005

We describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game. The algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation. While alternative techniques for computing the reachable set have been proposed, the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters. Because the time-dependent equation's solution is continuous and defined throughout the state space,...

A Viability Approach to Hamilton-Jacobi Equations: Application to Concave Highway Traffic Flux Functions

Aubin, J.-P.
Bayen, A.M.
Saint-Pierre, P.
2005

This paper presents a new approach which links the solution to a particular Hamilton-Jacobi partial differential equation to the solution of an optimal control problem provided by viability theory. It constructs the solution to this partial differential equation through its hypograph, which is defined as the capture basin of a target under an auxiliary dynamics that we define. The target itself represents the hypograph of a desired function. It is applied to concave Hamiltonian functions and has implications for the control of conservation laws with concave flux functions. It is a building...

Viability-Based Computations of Solutions to the Hamilton-Jacobi-Bellman Equation

Bayen, Alexandre M.
Claudel, Christian
Saint-Pierre, Patrick
Bemporad, Alberto
Bicchi, Antonio
2007

This article proposes a new capture basin algorithm for computing the numerical solution of a class of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) [3], based on a Lax-Hopf formula [2]. The capture basin algorithm is derived and implemented to perform numerical computations. Its performance is measured with highway data obtained for interstate I80 in California.

Traffic Flow Reconstruction Using Mobile Sensors and Loop Detector Data

Herrera, Juan C.
Bayen, Alexandre M.
2007

In order to develop efficient control strategies to improve traffic conditions on freeways, it is necessary to know the state of the freeway at any point in time and space. Using data collected from stationary detectors –such as loop detector stations– the density field can be currently reconstructed to a certain accuracy. Unfortunately, deploying this type of infrastructure is expensive, and its reliability varies. This article proposes and investigates new algorithms that make use of data provided by mobile sensors, in addition to that collected by stationary detectors, to reconstruct...

Identification de Paramètres pour les Equations de L'hydraulique à Surface Libre à L'aide de la Décomposition Modale

Wu, Qingfang
Amin, Saurabh
Munier, Simon
Bayen, Alexander M.
Litrico, Xavier
2007

Parameter identification for the shallow water equations using modal decomposition.

Computation of Solutions to the Moskowitz Hamilton-Jacobi-Bellman Equation Under Viability Constraints

Bayen, Alexandre M.
Claudel, Christian
Saint-Pierre, Patrick
2007

This article proposes a new capture basin algorithm for computing the numerical solution of a class of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs), based on a Lax-Hopf formula. The capture basin algorithm is derived and implemented to perform numerical computations of constrained solutions. The rate of convergence of this first order algorithm is assessed experimentally using an analytical benchmark problem. Finally, its performance is measured with highway data obtained for interstate 180 in California.

Parameter Identification for the Shallow Water Equation Using Modal Decomposition

Wu, Qingfang
Amin, Saurabh
Munier, Simon
Bayen, Alexandre M.
Litrico, Xavier
Belaud, Gilles
2007

A parameter identification problem for systems governed by first-order, linear hyperbolic partial differential equations subjected to periodic forcing is investigated. The problem is posed as a PDE constrained optimization problem with data of the problem given by the measured input and output variables at the boundary of the domain. By using the governing equations in the frequency domain, a spatially dependent transfer matrix relating the input variables to the output variables is obtained. It is shown that by considering a finite number of dominant oscillatory modes of the input, an...

On Stability of Switched Linear Hyperbolic Conservation Laws with Reflecting Boundaries

Amin, Saurabh
Hante, Falk M.
Bayen, Alexandre M.
Egerstedt, Magnus
Mishra, Bud
2008

We consider stability of an infinite dimensional switching system, posed as a system of linear hyperbolic partial differential equations (PDEs) with reflecting boundaries, where the system parameters and the boundary conditions switch in time. Asymptotic stability of the solution for arbitrary switching is proved under commutativity of the advective velocity matrices and a joint spectral radius condition involving the boundary data.

Solutions to Switched Hamilton-Jacobi Equations and Conservation Laws Using Hybrid Components

Claudel, Christian G.
Bayen, Alexandre M.
Egerstedt, Magnus
Mishra, Bud
2008

We investigate a class of hybrid systems driven by partial differential equations for which the infinite dimensional state can switch in time and in space at the same time. We consider a particular class of such problems (switched Hamilton-Jacobi equations) and define hybrid components as building blocks of hybrid solutions to such problems, using viability theory. We derive sufficient conditions for well-posedness of such problems, and use a generalized Lax-Hopf formula to compute these solutions. We illustrate the results with three examples: the computation of the hybrid components of a...

A Framework for Analyzing the Sensitivity of Traffic Data Quality to Sensor Location and Spacing: 15th World Congress on Intelligent Transport Systems and ITS America Annual Meeting 2008

Margulici, J.D.
Ban, Xuegang (Jeff)
Bayen, Alexander M.
Chu, Lianyu
2008

This paper presents a framework and tools developed to study the sensitivity of traffic data quality to detectors location and spacing. Our ultimate objective is to formulate generalized detector deployment guidelines that are based on the functional needs of practitioners, and for which funding can be objectively justified. Our approach consists in using trajectory sets obtained from field experiments and traffic simulation models as ground truth, and to run a traffic detector model from which we extract information that would normally be available to practitioners. Ground truth...