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

Bucci, F.

doi:10.4064/am35-3-4

Cited by 11 publications

(28 citation statements)

References 21 publications

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“…Model-specific analyses needs to be carried out on diverse PDE problems, in order to establish the regularity of boundary traces that is required in order to apply the introduced abstract theory (of the LQ-problem on a finite time interval). This has been achieved, indeed, in the case of boundary control problems for acoustic-structure and fluid-structure interactions as well; see [15] and [17,18], respectively.…”

Section: Introductionmentioning

confidence: 84%

A Theory of the Infinite Horizon LQ-Problem for Composite Systems of PDEs with Boundary Control

Acquistapace¹,

Bucci²,

Lasiecka³

2013

SIAM J. Math. Anal.

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We study the infinite horizon Linear-Quadratic problem and the associated algebraic Riccati equations for systems with unbounded control actions. The operator-theoretic context is motivated by composite systems of Partial Differential Equations (PDE) with boundary or point control. Specific focus is placed on systems of coupled hyperbolic/parabolic PDE with an overall 'predominant' hyperbolic character, such as, e.g., some models for thermoelastic or fluid-structure interactions. While unbounded control actions lead to Riccati equations with unbounded (operator) coefficients, unlike the parabolic case solvability of these equations becomes a major issue, owing to the lack of sufficient regularity of the solutions to the composite dynamics. In the present case, even the more general theory appealing to estimates of the singularity displayed by the kernel which occurs in the integral representation of the solution to the control system fails. A novel framework which embodies possible hyperbolic components of the dynamics has been introduced by the authors in 2005, and a full theory of the LQ-problem on a finite time horizon has been developed. The present paper provides the infinite time horizon theory, culminating in well-posedness of the corresponding (algebraic) Riccati equations. New technical challenges are encountered and new tools are needed, especially in order to pinpoint the differentiability of the optimal solution. The theory is illustrated by means of a boundary control problem arising in thermoelasticity.

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Section: Introductionmentioning

confidence: 84%

A Theory of the Infinite Horizon LQ-Problem for Composite Systems of PDEs with Boundary Control

Acquistapace¹,

Bucci²,

Lasiecka³

2013

SIAM J. Math. Anal.

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“…• regularity of the operator B * e A * • regularity of ∂θ ∂ν ∂Γ0 , • regularity of the operator B * e A * • A * ǫ regularity of ∂θt ∂ν ∂Γ0 . Precisely, Theorem 2.3 in [11] yields the sought decomposition (1.5) of the operator B * e A * t , along with the validity of 2., 3a) and 3c) of the Assumptions 1.4. A reworking on the proof of the singular estimate…”

Section: Verification Of the Basicmentioning

confidence: 83%

“…Theorem 1.9, shows that this is indeed the case, provided that the parameters γ and ǫ which occur in the Assumptions 1.4 fulfil ǫ < 1 − γ. This relation is feasible: to support the assertion, we revisit the actual values of γ and ǫ brought about by the (model-specific) trace regularity results established in [13], [2], [11], [12] in regard to distinct PDE problems, in the process of establishing that these fall into the present underlying framework.…”

Section: Introductionmentioning

confidence: 84%

“…We recall and discuss briefly three distinct PDE problems describing distinct physical interactions: from mechanical-thermal and acoustic-structure ones (in 2D and 3D, respectively) to fluid-elasticity ones in 3D. Each of these illustrations has been previously shown to fall within the class of abstract control systems defined by the Assumptions 1.4; see the papers [2], [11], [12], respectively. The values of γ and ǫ specifically suited for either case had explicitly arisen during the proofs of the soughtafter (respective) trace regularity results that pertain to a certain component of the solution to the uncontrolled PDE problem.…”

Section: The ǫ-γ Constraint: Pde Illustrationsmentioning

confidence: 99%

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On the infinitesimal generator of an optimal state semigroup

Acquistapace¹,

Bucci²

2021

Preprint

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In this article we fully describe the domain of the infinitesimal generator of the optimal state semigroup which arises in the theory of the linear-quadratic problem for a specific class of boundary control systems. This represents an improvement over earlier work of the authors, joint with Lasiecka, where a set inclusion was established, but not an equality. The key part of the proof of this result developes through appropriate asymptotic estimates that take advantage of the regularity analysis previously performed in the study of the optimization problem, while the powers of positive operators, interpolation and semigroups provide the analytical tools. We also attest to the validity of an assumed relation between two significant parameters in the case of distinct systems of coupled hyperbolic-parabolic partial differential equations which are pertinent to the underlying framework.

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“…The project carried out and accomplished in the studies [1,3] (with a few additions and refinements that are forthcoming) provides a framework for optimal control problems with quadratic functionals for a wider class of PDE systems which comprise both hyperbolic and parabolic components, with the latter subjected to boundary/interface control. This class has proven to be sufficiently general to encompass widely different PDE models such as certain thermoelastic systems, acoustic-structure and fluid-structure interactions (see [2], [12], [13,14]). The achievements of [1,3] include…”

Section: )mentioning

confidence: 99%

Improved boundary regularity for a Stokes-Lamé system

Bucci¹

2022

EECT

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This paper recalls a partial differential equations system, which is the linearization of a recognized fluid-elasticity interaction three-dimensional model. A collection of regularity results for the traces of the fluid variable on the interface between the body and the fluid is established, in the case a suitable boundary dissipation is present. These regularity estimates are geared toward ensuring the well-posedness of the Riccati equations which arise from the associated optimal boundary control problems on a finite as well as infinite time horizon. The theory of operator semigroups and interpolation provide the main tools.

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

Cited by 11 publications

References 21 publications

A Theory of the Infinite Horizon LQ-Problem for Composite Systems of PDEs with Boundary Control

A Theory of the Infinite Horizon LQ-Problem for Composite Systems of PDEs with Boundary Control

On the infinitesimal generator of an optimal state semigroup

Improved boundary regularity for a Stokes-Lamé system

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