Riesz Basis Property and Stability of Planar Networks of Controlled Strings

Han, Zhong‐Jie; Wang, Lei

doi:10.1007/s10440-009-9459-8

Cited by 14 publications

(19 citation statements)

References 24 publications

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“…Note that we can replace the Neumann condition (9) for the temperature at all a k 2 V int , with I te .a k / containing at least two elements, by the continuity condition [21],…”

Section: F Shelmentioning

confidence: 99%

“…In [7], the authors studied, in particular, the stabilization of a chain of beams and strings. See also [8][9][10][11].Another type of stabilization of an elastic material is to add thermoelastic materials to it. In [12][13][14], the authors proved that the system is then exponentially stable; see also [15] where the authors considered the case of beams and proved that the whole system is also exponentially stable.…”

mentioning

confidence: 99%

“…In [7], the authors studied, in particular, the stabilization of a chain of beams and strings. See also [8][9][10][11].…”

mentioning

confidence: 99%

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Exponential stability of a network of elastic and thermoelastic materials

Shel

2012

Math Methods in App Sciences

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The wave equation on an elastic body is conservative; to make the system stable, several authors have introduced different types of dissipative mechanisms, for example, a frictional damping [1] or frictional boundary conditions [2,3]. For the stabilization of a network governed by wave equations, we refer to [4,5], where the authors considered a star-shaped and tree-shaped networks of elastic strings, and they proved that when a feedback is applied on particular nodes, then the system will be polynomially stable but not exponentially stable. We can see, in [6], that the authors considered a network with delay term in the nodal feedbacks. In [7], the authors studied, in particular, the stabilization of a chain of beams and strings. See also [8][9][10][11].Another type of stabilization of an elastic material is to add thermoelastic materials to it. In [12][13][14], the authors proved that the system is then exponentially stable; see also [15] where the authors considered the case of beams and proved that the whole system is also exponentially stable. We want to know if this result holds true for a network of elastic and thermoelastic materials. To our knowledge, the asymptotic behavior of such a system has not been studied yet. In this paper, we consider particular cases of such network that can be partially generalized. In the first case, we suppose that two elastic edges cannot be adjacent (Figure 1). In the second one, we consider a tree of elastic materials, the leaves of which thermoelastic materials are added as follows: the thermoelastic body is related to only one leaf by an end, and the second is free or connects two leaves, with the condition that each leaf is connected to only one thermoelastic body (Figure 2). With the continuity condition for the displacement and the Neumann condition for the temperature at the internal nodes, we prove that the thermal effect is strong enough to stabilize the system. Similar to [16], we will use a frequency method as described in [17,18] (see also [19]).Before going on to the next discussion, let us first recall some definitions and notations about a 1-D network that will be used in this paper. We refer to [10,16,20] for more details.Let G be a network in the Euclidean space R m , with n vertices V D fa 1

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“…Note that we can replace the Neumann condition (9) for the temperature at all a k 2 V int , with I te .a k / containing at least two elements, by the continuity condition [21],…”

Section: F Shelmentioning

confidence: 99%

mentioning

confidence: 99%

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Exponential stability of a network of elastic and thermoelastic materials

Shel

2012

Math Methods in App Sciences

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“…In [20], the authors proved the exponential stability of a star-shaped network of beams with controls applied at external ends. In [9], the authors have proved the asymptotic stability of a star-shaped network of Timoshenko Beams. Ammari et al considered in [4] a chain of Euler-Bernoulli beams and strings; they establish some results of polynomial stability of such system.…”

Section: Introductionmentioning

confidence: 99%

Exponential Stability of a Network of Beams

Shel

2014

J Dyn Control Syst

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In this paper, we consider a network of elastic and thermoelastic Euler-Bernoulli beams. Our main result is to show that under certain conditions, the thermoelastic dissipation over some edges is sufficient to stabilize the whole system. Precisely, we establish the exponential stability of the network.Keywords Exponential stability · Frequency domain method · Network · Euler-Bernoulli beams · Elasticity and thermoelasticity Mathematics Subject Classifications (2010) 35B40 · 35M33 · 93D20.

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“…There also have been some results on the networks of the elastic systems. We refer to the early works [18,19] as well as the recent development on the networks of strings and beams [20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Stabilization and SDG condition of serially connected vibrating strings system with discontinuous displacement

Han

2012

Asian Journal of Control

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Based on joint feedback controls, the stabilization issue of a serially connected vibrating strings system is presented. Suppose that the two ends of the strings system are clamped and the vertical force of strings is continuous at interior nodes while the transversal displacement is discontinuous. The vertical force of strings at interior nodes are observed and the compensators are derived from the observation values. Then the feedback controllers are arranged at interior nodes to stabilize this system. Through a detailed spectral analysis, this closed loop system is proved to be asymptotically stable and there exists a sequence of (generalized) eigenvectors of the system that forms a Riesz basis with parentheses for the state space under certain conditions. Hence the spectrum-determined-growth (SDG) condition holds.

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Riesz Basis Property and Stability of Planar Networks of Controlled Strings

Cited by 14 publications

References 24 publications

Exponential stability of a network of elastic and thermoelastic materials

Exponential stability of a network of elastic and thermoelastic materials

Exponential Stability of a Network of Beams

Stabilization and SDG condition of serially connected vibrating strings system with discontinuous displacement

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