Improved observer design for heat equation with constant measurement delay via Legendre polynomials

In this paper, we present improved results on observer design for 1D heat equation. We first introduce an observer under delayed spatially point measurements that leads to an estimation error with time-delay. Inspired by recent developments in the area of delayed ODEs, we suggest augmented Lyapunov functionals based on the Legendre polynomials. Then, sufficient exponential stability conditions are derived in the form of linear matrix inequalities (LMIs) that are parameterized by the degree of the polynomials. Finally, a numerical example illustrates the efficiency of the results that allow to enlarge the value of delay preserving the stability by more than 20%.