Boundary exponential stabilization of a time-delay ODE-KdV cascaded system

This paper deals with the well-posedness and exponential stabilization problems of a cascaded control system consisting of a linear ordinary differential equation (ODE) and the one-dimensional linear Korteweg–de Vries (KdV) partial differential equation posed on a bounded interval $[0,l]$, where the states are subject to an arbitrary constant delay. The control input for the whole system acts at the left boundary of the KdV domain by Dirichlet condition, whereas the right boundary injects a Dirichlet term in the ODE subsystem. Based on the infinite dimensional backstepping method for the delay-free case, an explicit feedback control law is constructed. Under this feedback, we prove the well-posedness of the considered system in Hilbert space $H=R^n\times L^2(0,l)$ by using semigroup theory and its exponential stability in the topology of ${\Vert \cdot \Vert }_H$-norm by combining Lyapunov method with Halanay’s inequality and linear matrix inequalities (LMIs). A numerical example is provided to illustrate the result.