Distributed Parameter State Estimation for the Gray–Scott Reaction-Diffusion Model

Petro Feketa,Thomas Meurer,Alexander Schaum

doi:10.3390/systems9040071

Abstract

A constructive approach is provided for the reconstruction of stationary and non-stationary patterns in the one-dimensional Gray-Scott model, utilizing measurements of the system state at a finite number of locations. Relations between the parameters of the model and the density of the sensor locations are derived that ensure the exponential convergence of the estimated state to the original one. The designed observer is capable of tracking a variety of complex spatiotemporal behaviors and self-replicating patterns. The theoretical findings are illustrated in particular numerical case studies. The results of the paper can be used for the synchronization analysis of the master–slave configuration of two identical Gray–Scott models coupled via a finite number of spatial points and can also be exploited for the purposes of feedback control applications in which the complete state information is required.

Highlights

This paper aims at reconstructing stationary patterns and track spatiotemporal behavior of the one-dimensional Gray–Scott model utilizing measurements at a finite number of spatial locations
Complete state feedback of reaction-diffusion systems are often used for the control of semiconductor nanostructures [38] and particular examples of the control of patterns in the Gray–Scott model via delayed state feedback can be found in [21]
Inequalities (16a) and (16b) can be reformulated as sufficient conditions for the synchronization of the master-slave configuration of two identical 1D Gray–Scott models coupled via a finite number of spatial locations

Summary

Introduction

The Gray–Scott model [1,2], which is a simple prototype for models of complex isothermal autocatalytic reactions, is governed by a pair of coupled reaction–diffusion equations. This paper aims at reconstructing stationary patterns and track spatiotemporal behavior of the one-dimensional Gray–Scott model utilizing measurements at a finite number of spatial locations This task will be carried out by designing an observer, i.e., an auxiliary PDE system whose state asymptotically converges to the state of the original system in an appropriate norm as time goes to infinity. The idea of the PWI scheme for distributed parameter systems has been developed in [28,29,30,31], where the sensor location is determined by a suitable detectability analysis in terms of the Lipschitz constant of the nonlinearity and the dominant eigenvalue of the linear diffusion–convection operator Following this approach, the current paper proposes an exponentially convergent observer by direct injection of measurements into the observer dynamics at the measurement points.

Available Measurements and Main Results

Observer Setup

Proof of Theorem 1

Numerical Case Studies and Discussion

Findings

Conclusions and Outlook