Punctual control of a vibrating string: Numerical analysis

Bamberger, A.; Jaffré, Jérôme; Yvon, J. P.

doi:10.1016/0898-1221(78)90022-6

Cited by 20 publications

(13 citation statements)

References 4 publications

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“…We refer, e.g., to the work [2] by Bamberger, Ja re and Yvon who considered the scanning point feedback like…”

Section: Problem Formulation and Literaturementioning

confidence: 99%

“…Indeed, with (y 0 ; y 1 ) ∈ E the natural phase space for (4b) is E (which is our goal in this section, see Theorem 4.1 below) and hence y t lies in the space C([0; T ]; L 2 (0; 1)), whose elements may not necessarily have well-deÿned trace in L 2 (0; T ) on the curve s(·). We intend to show that this di culty can be overcome by employing the approach of the work [2], based on the explicit formula (similar to that by d'Alembert) for the solution of the associated open-loop problem:…”

Section: Open-loop Systemmentioning

confidence: 99%

“…where y (1) and y (2) are the restrictions of y, respectively, on Q T 1 and Q T 2 , and y t is deÿned on s(·) in Remark 4.3 below. Our physical justiÿcation for these somewhat 'controversial' notations is given in detail below in this section (see, in particular, Remark 4.1).…”

Section: Variational Problem Formulationmentioning

confidence: 99%

“…Assuming that ∈ L ∞ (0; T ), it was shown in Reference [2] that for any piecewise smooth curve x = (t); t ∈ (0; T ) such that…”

Section: Open-loop Systemmentioning

confidence: 99%

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Observability and stabilization of the vibrating string equipped with bouncing point sensors and actuators

Khapalov¹

2001

Math Methods in App Sciences

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SUMMARYThe issues of observability and stabilization are analysed for a vibrating string equipped, respectively, either with a point sensor measuring its displacement or a vertical tie-down viscous damper. It is well known that the aforementioned properties hold for the stationary devices of these types only when they are placed at the 'irrational points' of the string. The latter creates obvious di culties in applications. In this article we discuss how mobile point sensors and dampers, bouncing in a rather simple fashion between any two points on the string, can be employed instead in a more 'feasible and robust' way.

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“…We refer, e.g., to the work [2] by Bamberger, Ja re and Yvon who considered the scanning point feedback like…”

Section: Problem Formulation and Literaturementioning

confidence: 99%

Section: Open-loop Systemmentioning

confidence: 99%

Section: Variational Problem Formulationmentioning

confidence: 99%

“…Assuming that ∈ L ∞ (0; T ), it was shown in Reference [2] that for any piecewise smooth curve x = (t); t ∈ (0; T ) such that…”

Section: Open-loop Systemmentioning

confidence: 99%

See 2 more Smart Citations

Observability and stabilization of the vibrating string equipped with bouncing point sensors and actuators

Khapalov¹

2001

Math Methods in App Sciences

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“…To overcome this difficulty one may consider controls supported on moving points {γ(t)} 0≤t≤T , as suggested in [13] (the same strategy is analyzed in the context of feedback stabilization in [2]). The main advantage of moving controls is that it is easy to construct trajectories {γ(t)} 0≤t≤T for which the strategic property holds for γ(t) ∈ (0, L) a.e.…”

Section: Introductionmentioning

confidence: 99%