Control of a Time-Variant 1-D Linear Hyperbolic PDE using Infinite-Dimensional Backstepping

We derive a state-feedback controller for a scalar 1-D linear hyperbolic partial differential equation (PDE) with a spatially- and time-varying interior-domain parameter. The resulting controller ensures convergence to zero in a finite time d(1), corresponding to the propagation time from one boundary to the other. The control law requires predictions of the in-domain parameter a time d(1) into the future. The state-feedback controller is also combined with a boundary observer into an output-feedback control law. Lastly, under the assumption that the interior-domain parameter can be decoupled into a time-varying and a spatially-varying part, a stabilizing adaptive output-feedback control law is derived for an uncertain spatially varying parameter, stabilizing the system in the L-2-sense from a single boundary measurement only. All derived controllers are implemented and demonstrated in simulations.