Stabilization of linear 2 × 2 hyperbolic systems with uncertain coupling coefficients - part II: Swapping design

In this paper, we design a state feedback control law to adaptively stabilize a 2 × 2 linear hyperbolic system of partial differential equations (PDEs) with uncertain in-domain coupling coefficients. We do this using swapping design, where filters are designed so that the system states can be expressed as static, linear combinations of the filters and the uncertain parameters. Standard parameter identification methods can then be used to estimate the parameters. The adaptive observer is combined with a backstepping-based controller and boundedness in the ℒ₂-sense of all signals in the closed loop system, as well as integrability of the system states are shown. The theory is demonstrated in a simulation.