Output feedback boundary control of 2×2 semilinear hyperbolic systems

We consider the control and state estimation of a class of 2×2 semilinear hyperbolic systems with actuation and sensing collocated at one boundary. Our approach exploits the dynamics on the characteristic lines of the hyperbolic system. The control method using full-state feedback can be used for both stabilization of an equilibrium and tracking at an arbitrary point in the domain. The control objective is achieved globally in minimum time. A Lyapunov function is constructed to prove exponential convergence in the spatial supremum norm. For linear systems, the control input can be written explicitly as the inner product of kernels with the state, and turns out to be equivalent to the control input obtained from previously known backstepping methods. The observer achieves exact state estimation also in minimum time and, combined with the state-feedback controller, solves the output feedback control problem. The performance is demonstrated in a numerical example.