Delayed stabilization of parabolic PDEs via augmented Lyapunov functionals and Legendre polynomials

We first study stabilization of heat equation with globally Lipschitz nonlinearity. We consider the point measurements with constant delay and use spatial decomposition. Inspired by recent developments in the area of ordinary differential equations (ODEs) with time-delays, for the stability analysis, we suggest an augmented Lyapunov functional depending on the state derivative that is based on Legendre polynomials. Global exponential stability conditions are derived in terms of linear matrix inequalities (LMIs) that depend on the degree N of Legendre polynomials. The stability conditions form a hierarchy of LMIs: if the LMIs hold for N, they hold for N+1. The dual observer design problem with constant delay is also formulated. We further consider stabilization of Korteweg–de Vries–Burgers (KdVB) equation using the point measurements with constant delay. Due to the third-order partial derivative in KdVB equation, the Lyapunov functionals that depend on the state derivative are not applicable here, which is different from the case of heat equation. We suggest a novel augmented Lyapunov functional depending on the state only that leads to improved regional stability conditions in terms of LMIs. Finally, numerical examples illustrate the efficiency of the method.