Distributed stabilization of Korteweg–de Vries–Burgers equation in the presence of input delay

Abstract

We consider distributed stabilization of 1-D Korteweg–de Vries–Burgers (KdVB) equation in the presence of constant input delay. The delay may be uncertain, but bounded by a known upper bound. On the basis of spatially distributed (either point or averaged) measurements, we design a regionally stabilizing controller applied through distributed in space shape functions. The existing Lyapunov–Krasovskii functionals for heat equation that depend on the state derivative are not applicable to KdVB equation because of the third order partial derivative. We suggest a new Lyapunov–Krasovskii functional that leads to regional stability conditions of the closed-loop system in terms of linear matrix inequalities (LMIs). By solving these LMIs, an upper bound on the delay that preserves regional stability can be found, together with an estimate on the set of initial conditions starting from which the state trajectories of the system are exponentially converging to zero. This estimate includes a priori Lyapunov-based bounds on the solutions of the open-loop system on the initial time interval of the length of delay. Numerical examples illustrate the efficiency of the method.