scholarly journals Robustness of constant-delay predictor feedback for in-domain stabilization of reaction–diffusion PDEs with time- and spatially-varying input delays

Automatica ◽  
2021 ◽  
Vol 123 ◽  
pp. 109347
Author(s):  
Hugo Lhachemi ◽  
Christophe Prieur ◽  
Robert Shorten
Keyword(s):  
Reaction Diffusion ◽  
Constant Delay ◽  
Input Delays ◽  
Spatially Varying ◽  
Domain Stabilization

Extinction, stable pattern and their transition in a diffusive single species population with distributed maturity

2008 ◽  
Vol 19 (3) ◽  
pp. 285-309
Author(s):  
PEIXUAN WENG
Keyword(s):  
Distributed Delay ◽  
Reaction Diffusion ◽  
Single Species ◽  
Cylindrical Domain ◽  
Stable Pattern ◽  
Structured Population ◽  
Species Population ◽  
Single Species Population ◽  
Spatially Varying ◽  
Robin Boundary

We consider a single-species structured population with distributed maturity and spatial diffusion in a cylindrical domain subject to Neumann and Robin boundary conditions. We first establish the threshold property of the reaction–diffusion system with distributed delay and non-local interaction in a corresponding lower-dimensional domain, so that the system approaches either an extinction state or a stable spatially varying pattern. We then investigate the transition from the extinction state to the stable pattern of the system in the cylindrical domain.

Dead-Time Compensation for Wave/String PDEs

2011 ◽  
Vol 133 (3) ◽  
Author(s):  
Miroslav Krstic
Keyword(s):  
Reaction Diffusion ◽  
Equation System ◽  
Open Loop ◽  
Kernel Functions ◽  
Simple Relationship ◽  
Dramatic Form ◽  
Time Period ◽  
Dead Time Compensation ◽  
The Right ◽  
Input Delays

Smith predictorlike designs for compensation of arbitrarily long input delays are commonly available only for finite-dimensional systems. Only very few examples exist, where such compensation has been achieved for partial differential equation (PDE) systems, including our recent result for a parabolic (reaction-diffusion) PDE. In this paper, we address a more challenging wave PDE problem, where the difficulty is amplified by allowing all of this PDE’s eigenvalues to be a distance to the right of the imaginary axis. Antidamping (positive feedback) on the uncontrolled boundary induces this dramatic form of instability. We develop a design that compensates an arbitrarily long delay at the input of the boundary control system and achieves exponential stability in closed-loop. We derive explicit formulae for our controller’s gain kernel functions. They are related to the open-loop solutions of the antistable wave equation system over the time period of input delay (this simple relationship is the result of the design approach).

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