Ensembles of Hyperbolic PDEs: Stabilization by Backstepping

For the quite extensively developed partial differential equations (PDE) backstepping methodology for coupled linear hyperbolic PDEs, we provide a generalization from finite collections of such PDEs, whose states at each location in space are vector-valued, to previously unstudied infinite (continuum) ensembles of such hyperbolic PDEs, whose states are function-valued. The motivation for studying such systems comes from traffic applications (where driver and vehicle characteristics are continuously parametrized), fluid and structural applications, and future applications in population dynamics, including epidemiology. Our design is of an exponentially stabilizing scalar-valued control law for a PDE system in two independent dimensions, one spatial dimension and one ensemble dimension. In the process of generalizing PDE backstepping from finite to infinite collections of PDE systems, we generalize the results for PDE backstepping kernels to the continuously parametrized Goursat-form PDEs that govern such continuously parametrized kernels. The theory is illustrated with a simulation example, which is selected so that the kernels are explicitly solvable, to lend clarity and interpretability to the simulation results.