Output feedback boundary control of heterodirectional semilinear hyperbolic systems

Abstract

We solve the problem of stabilizing a general class of 1-d semilinear hyperbolic systems with an arbitrary number of states convecting in each direction and with the actuation and sensing restricted to one boundary. The control design is based on the dynamics on the characteristic lines along which the inputs propagate through the domain and the predictability of states in the interior of the domain up to the time they are affected by the inputs. In the context of broad solutions, the state-feedback controller drives systems with globally Lipschitz nonlinearities from an arbitrary initial condition to the origin in minimum time. Alternatively, it is possible to satisfy a tracking objective at the uncontrolled boundary or, for systems with C1-coefficients and initial conditions, to design the control inputs to obtain classical C1-solutions that also reach the origin in finite time. Further, we design an observer that estimates the distributed state from boundary measurements only. The observer combined with the state-feedback controller solves the output-feedback control problem.