Bilateral backstepping control of coupled linear parabolic PDEs with spatially varying coefficients

Kerschbaum, Simon; Deutscher, Joachim

doi:10.1016/j.automatica.2021.109923

Cited by 6 publications

(2 citation statements)

References 14 publications

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“…In the context of parabolic systems, significant progress has been obtained in exponential stabilization in L 2 -norm or H 1 -norm for linear parabolic PDEs by using the method of backstepping; see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. In general, when transformations are applied, the target systems are expected to be in a simple form, whose stability can be established easily.…”

Section: Introductionmentioning

confidence: 99%

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Exponential stabilization and continuous dependence of solutions on initial data in different norms for space-time-varying linear parabolic PDEs

Chen¹,

Zheng²,

Zhu³

2022

Preprint

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For an arbitrary parameter p ∈ [1, +∞], we consider the problem of exponential stabilization in the spatial L p -norm, and W 1,p -norm, respectively, for a class of anti-stable linear parabolic PDEs with space-time-varying coefficients in the absence of a Gevrey-like condition, which is often imposed on time-varying coefficients of PDEs and used to guarantee the existence of smooth (w.r.t. the time variable) kernel functions in the literature. Then, based on the obtained exponential stabilities, we show that the solution of the considered system depends continuously on the L p -norm, and W 1,p -norm, respectively, of the initial data. In order to obtain time-independent (and thus sufficiently smooth) kernel functions without a Gevrey-like condition and deal with singularities arising in the case of p ∈ [1, 2), we apply a combinatorial method, i.e., the combination of backstepping and approximation of Lyapunov functionals (ALFs), to stabilize the considered system and establish the continuous dependence of solutions on initial data in different norms.

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Section: Introductionmentioning

confidence: 99%

“…where λ is an arbitrary constant; see, e.g., [1,2,3,4,5,6,7,8,9,10,11]. In particular, for λ > 0, the stability in different norms of (1) has been well studied, and is used together with the inverse transformations to obtain the stability in L 2 -norm or H 1 -norm for the original parabolic PDEs.…”

Section: Introductionmentioning

confidence: 99%

Exponential stabilization and continuous dependence of solutions on initial data in different norms for space-time-varying linear parabolic PDEs

Chen¹,

Zheng²,

Zhu³

2022

Preprint

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PDE‐based containment control of linear multi‐agent systems

Deutscher,

Jung

2024

Intl J Robust & Nonlinear

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This contribution considers the containment control problem for linear multi‐agent systems (MAS) using continuum models in form of linear parabolic PDEs. For this, a networked controller is designed, that ensures the asymptotic convergence of finite‐dimensional agents into a containment area specified by dynamic leaders. This problem is traced back to an output regulation problem for a continuum model independent of the number of agents, which also takes disturbances into account. A prescribed formation is assigned for the MAS in the containment area by using Bézier curves. Using bilateral backstepping a tracking controller is designed, which allows to distribute the control effort between the two boundary agents. The states are estimated by an infinite‐dimensional disturbance observer. For this, both a folding observer with network communication and two boundary observers without network communication are introduced. A fully distributed solution of the containment control problem is obtained by employing a continuum signal model observer, which communicates the states of the leader and disturbance model to all agents. The desired communication topology is imposed after the design by a spatial discretization. A simulation example illustrates the effectiveness of the new method for the networked controller design.

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Integral input-to-state stabilization in different norms for a class of linear parabolic PDEs

Chen,

Zheng,

Zhu

2023

2023 35th Chinese Control and Decision Conference (CCDC)

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Bilateral backstepping control of coupled linear parabolic PDEs with spatially varying coefficients

Cited by 6 publications

References 14 publications

Exponential stabilization and continuous dependence of solutions on initial data in different norms for space-time-varying linear parabolic PDEs

Exponential stabilization and continuous dependence of solutions on initial data in different norms for space-time-varying linear parabolic PDEs

PDE‐based containment control of linear multi‐agent systems

Integral input-to-state stabilization in different norms for a class of linear parabolic PDEs

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