Sampled-data finite-dimensional observer-based control of 1-D Burgers’ equation

In this paper, we are concerned with the regional exponential stabilization of a 1-D Burgers’ equation under non-local/Neumann actuation and boundary measurement via a modal decomposition method. For non-local actuation, we suggest two control strategies: continuous-time control, and delayed sampled-data control implemented by zero-order hold (ZOH) device, both relying on finite-dimensional observer. For boundary actuation, we employ dynamic extension and consider an observer-based delayed sampled-data controller implemented by generalized hold device. For both cases, we suggest a direct Lyapunov method for the $H^1$-stability of the full-order closed-loop system. We provide efficient linear matrix inequality (LMI) conditions for finding the observer dimension, as well as upper bounds on the domain of attraction, sampling intervals and delays, that preserve the exponential stability. We prove that for some fixed upper bounds on the initial values and sampling intervals, the feasibility of LMIs for some $N$ (dimension of the observer) implies their feasibility for $N+1$. Numerical examples illustrate the efficiency of the proposed method.