Adaptive boundary control for heat equations with uncertain control coefficient and boundary disturbance

The adaptive state-feedback boundary control design is investigated for a class of heat equations with uncertain control coefficient and boundary disturbance. By choosing appropriate control Lyapunov function and parameter updating law, the adaptive controller is explicitly obtained, which only needs the measurement at the boundary of the system and guarantees that the original system state is L
2
[0, 1] stable (see the definition at the footnote of the page), and particularly, the state converges to zero when the boundary disturbance vanishes. It is worth emphasizing that, by skillfully choosing the initial condition of parameter updating law, the restriction on the initial condition of the system is moderately relaxed, which is usually described by the so-called compatible condition in the existing literature. A simulation example is presented to illustrate the effectiveness of the proposed method.