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

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 Lighthill-Whitham-Richards equation; a velocity control policy for a highway system; a data assimilation problem using Lagrangian measurements generated from NGSIM traffic data.