Internal stabilization of three interconnected semilinear reaction-diffusion PDEs with one actuated state

Constantinos Kitsos*, Rami Katz, Emilia Fridman

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


This work deals with the exponential stabilization of a system of three semilinear parabolic partial differential equations (PDEs), written in a strict feedforward form. The diffusion coefficients are considered distinct and the PDEs are interconnected via both a reaction matrix and a nonlinearity. Only one of the PDEs is assumed to be controlled internally, thereby leading to an underactuated system. Constructive and efficient control of such underactuated systems is a nontrivial open problem, which has been solved recently for the linear case. In this work, these results are extended to the semilinear case, which is highly challenging due the coupling introduced by the semilinearity. Modal decomposition is employed, where due to the semilinearity, the finite-dimensional part of the solution is coupled with the infinite-dimensional tail. A transformation is then employed to map the finite-dimensional part into a target system, which allows for an efficient design of a static linear proportional state-feedback controller. Furthermore, a high-gain approach is employed in order to compensate for the semilinear terms. Lyapunov stability analysis is performed, leading to LMI conditions guaranteeing exponential stability with an arbitrary decay rate. The LMIs are shown to always be feasible, provided the number of actuators and the value of the high gain parameter are large enough. Numerical examples demonstrate the proposed approach.

Original languageEnglish
Title of host publicationIFAC-PapersOnLine
EditorsHideaki Ishii, Yoshio Ebihara, Jun-ichi Imura, Masaki Yamakita
PublisherElsevier B.V.
Number of pages6
ISBN (Electronic)9781713872344
StatePublished - 1 Jul 2023
Event22nd IFAC World Congress - Yokohama, Japan
Duration: 9 Jul 202314 Jul 2023

Publication series

ISSN (Electronic)2405-8963


Conference22nd IFAC World Congress


  • Lyapunov stabilization
  • modal decomposition
  • semilinear parabolic PDE systems
  • underactuated systems


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