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This article proposes a new approach for computing a semi-explicit form of the solution to a class of Hamilton-Jacobi (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.Index Terms-Hamilton-Jacobi (HJ), next generation simulation (NGSIM), partial differential equations (PDEs).

This article presents a new method for explicitly computing solutions to a Hamilton-Jacobi partial differential equation for which initial, boundary and internal conditions are prescribed as piecewise affine functions. Based on viability theory, a Lax-Hopf formula is used to construct analytical solutions for the individual contribution of each affine condition to the solution of the problem. The results are assembled into a Lax-Hopf algorithm which can be used to compute the solution to the partial differential equation at any arbitrary time at no other cost than evaluating a semi-analytical expression numerically. The method being semi-analytical, it performs at machine accuracy (compared to the discretization error inherent to finite difference schemes). The performance of the method is assessed with benchmark analytical examples. The running time of the algorithm is compared with the running time of a Godunov scheme. Index Terms-Hamilton-Jacobi (HJ), initial conditions (ICs), Lax-Hopf formula, partial differential equation (PDE), piecewise affine (PWA), terminal conditions (TCs).

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This article proposes a new method for data assimilation and data reconciliation problems applicable to systems modeled by conservation laws. The problem is solved directly in the equivalent format of a Hamilton-Jacobi partial differential equation, for which the solution is fully characterized by a Lax-Hopf formula. Using properties of the solution, we prove that when the data of the problem is prescribed in piecewise affine form, the resulting constraints which consist of the partial differential equation in data assimilation and reconciliation problems are convex, and can be instantiated explicitly. This property enables us to identify a class of data assimilation and data reconciliation problems that can be formulated using convex programs in standard form. We illustrate the capabilities of the method for reconstruction of highway traffic flow using experimental data generated from the Mobile Century experiment.

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