Input-to-State Stabilization in H1-Norm for Boundary Controlled Linear Hyperbolic PDEs with Application to Quantized Control

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. For this system class, the problem of designing dynamic controllers for input-to-state stabilization in H-1-norm with respect to measurement errors is considered. 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.