Concurrent learning adaptive stabilization of 2 × 2 linear hyperbolic PDE systems with boundary parameter uncertainty

Existing results on adaptive boundary control for hyperbolic PDE systems based on regular gradient descent or least squares laws have two general limitations, namely, there is no theoretical guarantee for the convergence of parameter estimation error and only asymptotic system stability is achieved. This paper proposes a novel adaptive controller for stabilizing 2 × 2 hyperbolic PDE systems with uncertain distal refection. As only the boundary state is measurable, an adaptive observer is designed to estimate the system states and the boundary uncertainty, simultaneously. For the boundary uncertainty, we develop a new parameter estimation law, called the concurrent learning parameter estimation law, that simultaneously leverages the real-time and historical system states to update the estimated parameter. It is shown that the estimation errors of the system states and the uncertain parameter are exponentially convergent. Then, an adaptive output-feedback backstepping controller is developed based on the observed states and the estimated parameter. Thanks to the accurate parameter estimation, the controller guarantees exponential system stability rather than asymptotic stability. Finally, the proposed control scheme is demonstrated in a numerical comparison simulation.