Adaptive Stabilization of $ 2 \times 2$ Linear Hyperbolic Systems With an Unknown Boundary Parameter From Collocated Sensing and Control

In this paper, we solve an adaptive control problem for a class of 2 × 2 linear hyperbolic partial differential equations, where sensing and actuation are restricted to the boundary anticollocated with an uncertain parameter. This is done by combining a recently derived adaptive observer for the system states and the uncertain parameter, with an adaptive control law. Proof of L2-boundedness for all signals in the closed loop is given, and the system states are proved to converge to zero pointwise in space. The theory is demonstrated in a simulation.