Stabilization Using In-domain Actuator: A Numerical Method for a Non Linear Parabolic Partial Differential Equation

Abstract

This paper deals with the problem of null controllability for an unstable nonlinear parabolic partial differential equation (PDE) system considering in-domain actuator. The main objective of this communication is to provide an efficient control law in order to stabilize the system state close to zero in a desired time whatever the initial state is. A numerical approach is developed and in order to highlight the relevance of the proposed control strategy, a realistic physical problem is investigated. Thermal evolution of a thin rod with homogeneous Dirichlet boundaries conditions is considered. Thermal state is described by the heat equation and assuming that thermal conductivity is temperature dependent, a nonlinear mathematical model has to be taken into account. Considering that all the model inputs are known, a direct problem is numerically solved (regarding a finite element method) in order to estimate the temperature at each point of the 1D geometry and at each instant. Then an inverse problem is formulated in such a way as to determine the in-domain control which ensures a final temperature close to zero. An iterative regularization method based on the conjugate gradient method (CGM) is developed for the minimization of a quadratic cost function (output error). Several numerical experimentations are provided in order to discuss the numerical approach attractiveness.

Quasi-Online Disturbance Rejection for Nonlinear Parabolic PDE using a Receding Time Horizon Control

“…Optimization algorithm is briefly described as follows in order to minimize criterion (2) (in [8] a similar algorithm is presented in offline context):…”

“…In the following the method presented in [8] is modified in order to provide a quasi-online numerical strategy for the identification of control laws. The goal is to identify the control that keeps the system state close to zero (null controllability) using the conjugate gradient problem.…”