Nonlinear Feedback Control of Parabolic Partial Differential Equation Systems with Time-dependent Spatial Domains

Abstract

This paper proposes a methodology for the synthesis of nonlinear finite-dimensional time-varying output feedback controllers for systems of quasi-linear parabolic partial differential equations (PDEs) with time-dependent spatial domains, whose dynamics can be partitioned into slow and fast ones. Initially, a nonlinear model reduction scheme, similar to the one introduced in Christofides and Daoutidis, J. Math. Anal. Appl. 216 (1997), 398–420, which is based on combinations of Galerkin's method with the concept of approximate inertial manifold is employed for the derivation of low-order ordinary differential equation (ODE) systems that yield solutions which are close, up to a desired accuracy, to the ones of the PDE system, for almost all times. Then, these ODE systems are used as the basis for the explicit construction of nonlinear time-varying output feedback controllers via geometric control methods. The controllers guarantee stability and enforce the output of the closed-loop parabolic PDE system to follow, up to a desired accuracy, a prespecified response for almost all times, provided that the separation of the slow and fast dynamics is sufficiently large. Differences in the nature of the model reduction and control problems between parabolic PDE systems with fixed and moving spatial domains are identified and discussed. The proposed control method is used to stabilize an unstable steady state of a diffusion-reaction process whose spatial domain changes with time. It is shown to lead to a significant reduction on the order of the stabilizing nonlinear output feedback controller and outperform a nonlinear controller synthesis method that does not account for the variation of the spatial domain.