Uniform stability in structural acoustic models with flexible curved walls

Cagnol, John; Lasiecka, Irena; Lebiedzik, Catherine; Zolésio, Jean-Paul

doi:10.1016/s0022-0396(02)00029-3

Cited by 21 publications

(33 citation statements)

References 18 publications

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“…The LQ-problem for a structural acoustic model with curved-rather than flatflexible wall Γ0 has been explored in [10]; for more on modeling and control of shell-like structures, see the same authors' work in the references therein.…”

Section: Statement Of the Problem And Main Resultsmentioning

confidence: 99%

Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results

Bucci

2008

Appl. Math. (Warsaw)

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Abstract. We consider a structural acoustic problem with the flexible wall modeled by a thermoelastic plate, subject to Dirichlet boundary control in the thermal component. We establish sharp regularity results for the traces of the thermal variable on the boundary in case the system is supplemented with clamped mechanical boundary conditions. These regularity estimates are most crucial for validity of the optimal control theory developed by Acquistapace et al. [Adv. Differential Equations, 2005], which ensures well-posedness of the corresponding differential Riccati equations. The proof takes full advantage of the exceptional boundary regularity of the mechanical component of the clamped thermoelastic system as well as of the sharp trace theory pertaining to wave equations with Neumann boundary data.1. Introduction. This paper continues-and concludes-the study initiated in [7], focused on a class of boundary control problems for a system of partial differential equations (PDE) describing fluid-structure interactions (structural acoustic model), which also include thermal effects. Our primary goal is to discuss the question of solvability of the associated quadratic optimal control problems over a finite time interval, along with well-posedness of the corresponding differential Riccati equations (DRE). As it is known and will become clearer later, this naturally leads us to undertake a preliminary investigation of the regularity properties of the solution to a dual (homogeneous) boundary value problem.The structural acoustic model under investigation is the same as in [7], except for the boundary conditions. More precisely, the PDE system (that 2000 Mathematics Subject Classification: 35B37, 35M20, 35B65, 49J20, 93C20.

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Section: Statement Of the Problem And Main Resultsmentioning

confidence: 99%

Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results

Bucci

2008

Appl. Math. (Warsaw)

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“…A 2-dimensional thermoelastic plate with clamped boundary conditions and subject to interior point control -the model of the present paper -is an ideal wall of a structural acoustic chamber, subject to piezo-ceramic control action, for the purpose of noise reduction [1,2,3,6,15,22,4,5] to quote a few references.…”

Section: The Pathology Of the Critical Case N =mentioning

confidence: 99%

The Optimal Interior Regularity for the Critical Case of a Clamped Thermoelastic System with Point Control Revisited

Lebiedzik

Triggiani

2011

Modern Aspects of the Theory of Partial Differential Equations

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Abstract. In the case of clamped thermoelastic systems with interior point control defined on a bounded domain Ω, the critical case is n = dim Ω = 2. Indeed, an optimal interior regularity theory was obtained in [Triggiani, Discrete Contin. Dyn. Syst., 2007] for n = 1 and n = 3. However, in this reference, an ' -loss' of interior regularity has occurred due to a peculiar pathology: the incompatibility of the B.C. of the spaces H 3/2 0 (Ω) and H 3/2 00 (Ω). This problem for n = 2 was rectified in [Triggiani, J. Differential Equations, 2008]: this establishes the sought-after interior regularity of the thermoelastic problem through a technical analysis based on sharp boundary (trace) regularity theory of Kirchhoff and wave equations. As an additional bonus, a sharp boundary regularity of the elastic displacement is also obtained. In the present paper, we revisit that problem using a technique developed by these authors to circumvent the pathology of the incompatible boundary conditions. This yields a more direct proof of the optimal interior regularity (but not of the boundary regularity).

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“…In [5], the full computation of the strain energy was carried out and the resulting model was coupled to a three-dimensional wave equation. In addition, the resulting structural-acoustic system was shown to be uniformly stable.…”

Section: Literaturementioning

confidence: 99%

“…In addition, the shell is shallow in the sense that the second fundamental form (here given in terms of the oriented distance function as D 2 b) and its derivative (D 3 b) are small. This assumption allows us to neglect certain terms in the strain energy in comparison to the model in [5,6] and yields the intrinsic version of the classical shallow shell equations. For a detailed justification of these assumptions we refer to Koiter [15,16].…”

Section: Model Hypothesesmentioning

confidence: 99%

See 1 more Smart Citation

Exact boundary controllability of a shallow intrinsic shell model

Lebiedzik

2007

Journal of Mathematical Analysis and Applications

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We consider a variant of a Koiter shell model based on the intrinsic geometry methods of Michael Delfour and Jean-Paul Zolésio. This model, derived in [J. Cagnol, I. Lasiecka, C. Lebiedzik, J.-P. Zolésio, Uniform stability in structural acoustic models with flexible curved walls, J. Differential Equations 186 (1) (2003) 88-121], relies heavily on the oriented distance function which describes the geometry. Here, we establish continuous observability estimates in the Dirichlet case with an explicit observability time, under an additional shallowness assumption and a checkable geometric condition. This yields (by duality) exact controllability for this class of intrinsically modelled shells.

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Uniform stability in structural acoustic models with flexible curved walls

Cited by 21 publications

References 18 publications

Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results

Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results

The Optimal Interior Regularity for the Critical Case of a Clamped Thermoelastic System with Point Control Revisited

Exact boundary controllability of a shallow intrinsic shell model

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