Polynomial and analytic stabilization of a wave equation coupled with an Euler–Bernoulli beam

Ammari, Kaïs; Nicaise, Serge

doi:10.1002/mma.1052

Cited by 8 publications

(10 citation statements)

References 16 publications

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“…If ω = ω n for some n ∈ I, then Re P (iω n ) > 0 and assumption (3) imply that S A (iω n ) is boundedly invertible. Thus {iω k } k∈I ⊂ ρ(A e ).…”

Section: Stability Of Coupled Passive Systemsmentioning

confidence: 99%

“…x e (t) = A e x e (t), x e (0) = x e0 ∈ X e (1. 3) on the Hilbert space X e = X × Z. The notation (A c , B c , C c , D c ) and our results on the closed-loop stability are motivated by the second part of the paper where we study robust output tracking and disturbance rejection for the system (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…Strong and exponential closed-loop stabilities of infinite-dimensional systems have been studied in the literature for passive one-dimensional boundary control systems [36,30], coupled systems with collocated inputs and outputs [15], and passive systems coupled with finite-dimensional systems [44]. Polynomial stability of coupled systems has been studied extensively in the context of coupled linear partial differential equations [3,1,6,2], and for abstract hyperbolic-parabolic systems [20].…”

Section: Introductionmentioning

confidence: 99%

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Stability and Robust Regulation of Passive Linear Systems

Paunonen¹

2019

SIAM J. Control Optim.

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We study the stability of coupled impedance passive regular linear systems under power-preserving interconnections. We present new conditions for strong, exponential, and non-uniform stability of the closed-loop system. We apply the stability results to the construction of passive error feedback controllers for robust output tracking and disturbance rejection for strongly stabilizable passive systems. In the case of nonsmooth reference and disturbance signals we present conditions for non-uniform rational and logarithmic rates of convergence of the output. The results are illustrated with examples on designing controllers for linear wave and heat equations, and on studying the stability of a system of coupled partial differential equations.2010 Mathematics Subject Classification. 93C05, 47D06, 93D20, 93B52 (47A10, 35B35, 93D15).

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“…If ω = ω n for some n ∈ I, then Re P (iω n ) > 0 and assumption (3) imply that S A (iω n ) is boundedly invertible. Thus {iω k } k∈I ⊂ ρ(A e ).…”

Section: Stability Of Coupled Passive Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stability and Robust Regulation of Passive Linear Systems

Paunonen¹

2019

SIAM J. Control Optim.

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“…Proof of the first assertion of the Theorem 2.1. For this proof, we need a result of the following lemma inspired by [19] (see also [7]).…”

Section: Some Background On a Class Of Dynamical Systemsmentioning

confidence: 99%

Polynomial and Analytic Boundary Feedback Stabilization of Square Plate

Nouira

2009

B Soc Paran Mat

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We consider a boundary feedback stabilization problem of the plate equation in a square, in the case where the geometric condition of Ammari-Tucsnak [6] is not satisfied. We prove a polynomial decay for regular initial data. Moreover, we prove an exponential stability result for some subspace of the energy space. Finally, we give a precise estimate on the analyticity of reachable functions where we have an exponential stability.

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“…The study on these distributed parameter networks has caught much attention in mathematics and engineering since 80's last century. Many nice results are available, covering the controllability, observability and stabilization of the networks (e.g., [1,3,7,9,11,13,14,17,19]). Most of them are concerned with 1-D wave and beam equation expanded on simple graphs, while a few focus on the complex networks which have a number of edges and complicated connections.…”

Section: Introductionmentioning

confidence: 99%

Controller design for bush-type 1-d wave networks

Zhang

2010

ESAIM: COCV

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Abstract. In this paper, we introduce a new method for feedback controller design for the complex distributed parameter networks governed by wave equations, which ensures the stability of the closed loop system. This method is based on the uniqueness theory of ordinary differential equations and cutting-edge approach in the graph theory, but it is not a simple extension. As a realization of this idea, we investigate a bush-type wave network. The well-posedness of the closed loop system is obtained via Lax-Milgram's lemma and semigroup theory. The validity of cutting-edge method is proved by spectral analysis approach. In particular, we give a detailed procedure of cutting-edge for the bush-type wave networks. The results show that if we impose feedback controllers, consisting of velocity and position terms, at all the boundary vertices and at most three velocity feedback controllers on the cycle, the system is asymptotically stabilized. Finally, some examples are given.Mathematics Subject Classification. 35L05, 35L90, 37L15, 93C20, 93D15.

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Polynomial and analytic stabilization of a wave equation coupled with an Euler–Bernoulli beam

Cited by 8 publications

References 16 publications

Stability and Robust Regulation of Passive Linear Systems

Stability and Robust Regulation of Passive Linear Systems

Polynomial and Analytic Boundary Feedback Stabilization of Square Plate

Controller design for bush-type 1-d wave networks

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