Dynamical observers for parabolic equations with spatial point measurements

TL;DR

This paper introduces an exponential Luenberger dynamical observer for estimating states of nonautonomous semilinear parabolic equations using finite spatial point measurements, with applications demonstrated through simulations.

Contribution

It develops an explicit output injection operator and a state space decomposition method for state estimation with spatial point sensors in parabolic PDEs.

Findings

Effective state estimation demonstrated on Kuramoto--Sivashinsky models.

Explicit output injection operator derived for finite point measurements.

Simulation results confirm the method's applicability to flame and fluid flow models.

Abstract

An exponential Luenberger dynamical observer is proposed to estimate the state of a general class of nonautonomous semilinear parabolic equations. The result can be applied to the case where the output is given by state measurements taken at a finite number of spatial points, that is, to the case where our sensors are a finite number of delta distributions. The output injection operator is explicit and the derivation of the main result involves the decomposition of the state space into a direct sum of two oblique components depending on the set of sensors. Simulations are presented as an application to the Kuramoto--Sivashinsky models for flame propagation and fluid flow.