Ensembles of Hyperbolic PDEs: Stabilization by Backstepping

TL;DR

This paper extends the PDE backstepping methodology from finite collections to infinite ensembles of hyperbolic PDEs, enabling stabilization in systems with continuous parameter variations, with applications in traffic, fluid, structural, and population dynamics.

Contribution

It introduces a novel generalization of PDE backstepping to infinite ensembles, including new kernel equations and an explicit control law for systems with continuum parameters.

Findings

Successfully generalized backstepping kernels to continuum systems.

Designed an exponentially stabilizing control law for a two-dimensional PDE system.

Provided a simulation example with explicit kernels demonstrating the theory.

Abstract

For the quite extensively developed 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.