Linear and Quadratic Programming Formulations of Data Assimilation or Data Reconciliation Problems for a Class of Hamilton-Jacobi Equations

This article proposes a new method for data assimilation and data reconciliation applicable to systems modeled by conservation laws. The state of the system is written in the form of a scalar Hamilton-Jacobi (HJ) partial differential equation (PDE), for which the solution is fully characterized by a LaxHopf formula. Using the properties of the solution, we prove that when the data of the problem is prescribed in piecewise affine form, the constraints of the model are in standard convex form, and can be computed 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.