Stabilization of a Heterodirectional Bradytachic (1D, 2D) PDE Pair

Abstract

A scalar hyperbolic PDE, actuated at one boundary, is coupled with a fast PDE in two spatial dimensions: transport in the axial coordinate and in an internal coordinate, and possibly diffusion in the internal coordinate. Backstepping designs exist for coupled hyperbolic systems, for their ensembles, and for their continua, but in all of these the second variable of the fast subsystem carries no transport or diffusion; when it does, no design is available, and from a scalar input no exact control of the two-dimensional state is to be expected. The pair is stabilized here by time-scale separation: in the quasi-steady limit the two-dimensional subsystem collapses into a spatial Volterra operator inside a one-dimensional reduced plant, backstepping applies there, and the two-dimensional subsystem is left to be a stable boundary layer. One theorem establishes exponential stability of the pair for every time-scale ratio below an explicit threshold, uniformly in the internal diffusion coefficient down to the pure-transport case.