Local exponential stabilization of Fisher’s equation using the backstepping technique

Yu, Xin; Xu, Chao; Chu, Jun

doi:10.1016/j.sysconle.2014.09.002

Cited by 21 publications

(39 citation statements)

References 18 publications

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“…We present first the well‐posedness of system , which can be transformed into an integrated equation by Duhamel's principle . Note that the well‐posedness of semilinear parabolic equations has been intensively addressed in the literature . However, in the considered setting, the nonhomogeneous term

\sum_{i = 1}^{n} δ u_{i} false(t false)

belongs to D ′ ( A ) rather than D ( A ).…”

Section: Problem Statement and Stability Analysismentioning

confidence: 99%

“…34 Note that the well-posedness of semilinear parabolic equations has been intensively addressed in the literature. 35,36 However, in the considered setting, the nonhomogeneous term ∑ n i=1 u i (t) belongs to D ′ (A) rather than D(A). Therefore, the existing results on the well-posedness of semilinear parabolic equations cannot be directly applied.…”

Section: Lemma 1 (See the Work Of Byrnes Et Al 33 )mentioning

confidence: 99%

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Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

Yang

Zheng

Zhu

2019

Intl J Robust & Nonlinear

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Summary This paper addresses the problem of asymptotic output tracking of a class of semilinear parabolic equations with pointwise in‐domain actuation. First, the assessment of the well‐posedness of the considered systems is performed, and then, the stability of boundary controlled systems is analyzed via Chaffee‐Infante equation and Fisher's equation. The application of the zero dynamics inverse design results in a dynamic control scheme that is implemented by using the technique of trajectory planning for flat systems and the Adomian decomposition method. The convergence of the solution of the original systems to that of the corresponding zero dynamics and the convergence of the solution expressed by an Adomian series are also analyzed. Numerical simulations are carried out to illustrate the effectiveness of the developed approach.

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\sum_{i = 1}^{n} δ u_{i} false(t false)

belongs to D ′ ( A ) rather than D ( A ).…”

Section: Problem Statement and Stability Analysismentioning

confidence: 99%

Section: Lemma 1 (See the Work Of Byrnes Et Al 33 )mentioning

confidence: 99%

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

Yang

Zheng

Zhu

2019

Intl J Robust & Nonlinear

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“…Later, in References and , the authors have introduced an invertible integral transformation that transforms the original parabolic PDE into an asymptotically stable one called target system. Recently, some nonlinear control PDEs are stabilized using the backstepping method . More recently, the backstepping method is extended to design a feedback control law for coupled PDE‐ODE .…”

Section: Introductionmentioning

confidence: 99%

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

Benabdallah

2020

Intl J Robust & Nonlinear

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This article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1 − d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite-dimensional backstepping design for state feedback stabilization design of coupled PDE-ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right-hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method. K E Y W O R D Sboundary stabilization, coupled partial differential equation-ordinary differential equation, infinite-dimensional backstepping, uncertain nonlinear system INTRODUCTIONIn the last decades, coupled partial differential equation (PDE)-ordinary differential equation (ODE) systems have attracted much attention in research communities because such systems can be used to model various processes such as road traffic, 1 gas flow pipeline, 2 power converters connected to transmission lines, 3 oil drilling, 4 and many other systems. The coupled PDE-ODE system considered in this article can be viewed as a perturbation of a simplified version that models the solid-gas interaction of heat diffusion and chemical reaction, where the interaction occurs at the interface. 5Abbreviations: ODE, ordinary differential equation; PDE, partial differential equation. Int J Robust Nonlinear Control. 2020;30:3023-3038. wileyonlinelibrary.com/journal/rnc

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“…Recently, X. Yu et al studied the local boundary feedback stabilizing of 1-D Fisher's equation by Backstepping method ( [13]), wherein it is proved the feedback controller stabilizes the system in both L 2 (0, 1) and H 1 (0, 1), and numerical examples are provided to illustrate the effectiveness. Fisher's equation is a nonlinear parabolic equation firstly proposed by Fisher to model the advance of a mutant gene in an infinite one-dimensional habitat [14].…”

Section: Introductionmentioning

confidence: 99%

“…In this work, for stabilizing the Fisher's equation, we shall adopt the technique in [9] to design a feedback controller, which is different from the backstepping method applied in [13], but the form is much simpler. In the next section, by following similar arguments as in [9], we shall prove that the designed finite dimensional boundary feedback controller globally stabilizes the linearized equation…”

Section: Introductionmentioning

confidence: 99%

Boundary feedback stabilization of Fisher’s equation

Liu

Munteanu

2016

Systems & Control Letters

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The aim of this work is to design an explicit finite dimensional boundary feedback controller for locally exponentially stabilizing the equilibrium solutions to Fisher's equation in both L 2 (0, 1) and H 1 (0, 1). The feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation. This stabilizing procedure is applicable for any level of instability, which extends the result of [2] for nonlinear parabolic equations. The effectiveness of the approach is illustrated by a numerical simulation.

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Local exponential stabilization of Fisher’s equation using the backstepping technique

Cited by 21 publications

References 18 publications

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

Boundary feedback stabilization of Fisher’s equation

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