Title: Disturbance-to-State Stabilization and Quantized Control for Linear Hyperbolic Systems,
Abstract: We consider a system of linear hyperbolic PDEs where the state at one of the
boundary points is controlled using the measurements of another boundary point.
Because of the disturbances in the measurement, the problem of designing
dynamic controllers is considered so that the closed-loop system is robust with
respect to measurement errors. Assuming that the disturbance is a locally
essentially bounded measurable function of time, we derive a
disturbance-to-state estimate which provides an upper bound on the maximum norm
of the state (with respect to the spatial variable) at each time in terms of
$\mathcal{L}^\infty$-norm of the disturbance up to that time. The analysis is
based on constructing a Lyapunov function for the closed-loop system, which
leads to controller synthesis and the conditions on system dynamics required
for stability. As an application of this stability notion, the problem of
quantized control for hyperbolic PDEs is considered where the measurements sent
to the controller are communicated using a quantizer of finite length. The
presence of quantizer yields practical stability only, and the ultimate bounds
on the norm of the state trajectory are also derived.