Global Stabilization of Nonlinear Finite Dimensional System with Dynamic Controller Governed by 1 − d Heat Equation with Neumann Interconnection

Mohsen Dlala and Abdallah Benabdallah
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Mohsen Dlala: Department of Mathematics, College of Sciences, Qassim University, Buraidah 51452, Saudi Arabia
Abdallah Benabdallah: Higher Institute of Computer Science and Multimedia, University of Sfax, Sfax 3021, Tunisia

Mathematics, 2022, vol. 10, issue 2, 1-15

Abstract: This paper deals with the stabilization of a class of uncertain nonlinear ordinary differential equations (ODEs) with a dynamic controller governed by a linear 1 − d heat partial differential equation (PDE). The control operates at one boundary of the domain of the heat controller, while at the other end of the boundary, a Neumann term is injected into the ODE plant. We achieve the desired global exponential stabilization goal by using a recent infinite-dimensional backstepping design for coupled PDE-ODE systems combined with a high-gain state feedback and domination approach. The stabilization result of the coupled system is established under two main restrictions: the first restriction concerns the particular classical form of our ODE, which contains, in addition to a controllable linear part, a second uncertain nonlinear part verifying a lower triangular linear growth condition. The second restriction concerns the length of the domain of the PDE which is restricted.

Keywords: coupled PDE-ODE; infinite dimensional backstepping; boundary stabilization; uncertain nonlinear system (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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