Sampled-data finite-dimensional boundary control for 1-D Burgers' equation

In this paper, we are concerned with the regional exponential stabilization of a 1-D Burgers' equation under Neumann actuation via modal decomposition method and dynamic extension. We consider a sampled-data finite-dimensional boundary control, which is implemented via a generalized hold device. We use Wirtinger-based piecewise continuous-time Lyapunov functional to compensate sampling of the finite-dimensional state, and provide the H¹-stability analysis for the full-order closed-loop system. Given a decay rate, we provide the efficient linear matrix inequality (LMI) conditions for finding the controller dimension and gain, as well as a bound on the domain of attraction. We prove that for some fixed upper bounds on the initial value and sampling intervals, the feasibility of LMIs for some N (dimension of the controller) implies their feasibility for N + 1. Numerical example illustrates the efficiency of the proposed method.