3,645 research outputs found
The critical case of clamped thermoelastic systems with interior point control: Optimal interior and boundary regularity results
AbstractIn the case of clamped thermoelastic systems with interior point control defined on a bounded domain Ω, the critical case is n=dimΩ=2. Indeed, an optimal interior regularity theory was obtained in [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993–1004] for n=1 and n=3. However, in this reference, an ‘ϵ-loss’ of interior regularity has occurred due to a peculiar pathology: the incompatibility of the B.C. of the spaces H032(Ω) and H0032(Ω). The present paper manages to establish that, indeed, one can take ϵ=0, thus obtaining an optimal interior regularity theory also for the case n=2. The elastic variables have the same interior regularity as in the corresponding elastic problem [R. Triggiani, Regularity with interior point control, Part II: Kirchhoff equations, J. Differential Equations 103 (1993) 394–421] (Kirchhoff). Unlike [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993–1004], the present paper 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. In the process, a new boundary regularity result, not contained in [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993–1004], is obtained for the elastic displacement of the thermoelastic system
On the lack of exact controllability for mild solutions in Banach spaces
AbstractIt is shown that exact controllability in finite time for linear control systems given on an infinite dimensional separable Banach space in integral form (mild solution) can never arise using locally L1-controls, if the operator through which the control acts on the system is compact. This improves a previous result of the author, by removing the assumption that the state space have a basis. It is suggested by the recent discovery that a separable Banach space need not have a basis
Mathematical analysis of PDE systems which govern °uid structure interactive phenomena
In this paper, we review and comment upon recently derived results for time dependent partial differential equation (PDE) models, which have been used to describe the various fluid-structure interactions which occur in nature. For these fluid-structure PDEs, this survey is particularly focused on the authors\u27 results of (i) semigroup wellposedness, (ii) stability, and (iii) backward uniqueness
Estimation with ultimate quantum precision of the transverse displacement between two photons via two-photon interference sampling measurements
We present a quantum sensing scheme achieving the ultimate quantum
sensitivity in the estimation of the transverse displacement between two
photons interfering at a balanced beam splitter, based on transverse-momentum
sampling measurements at the output. This scheme can possibly lead to enhanced
high-precision nanoscopic techniques, such as super-resolved single-molecule
localization microscopy with quantum dots, by circumventing the requirements in
standard direct imaging of cameras resolution at the diffraction limit, and of
highly magnifying objectives. Interestingly, the ultimate spatial precision in
nature is achieved irrespectively of the overlap of the two displaced photonic
wavepackets. This opens a new research paradigm based on the interface between
spatially resolved quantum interference and quantum-enhanced spatial
sensitivity.Comment: 13 pages, 4 figure
Fluid-Viscoelastic Structure Interaction
We consider a fluid- structure interaction model consisting of the N-S equations coupled with a system of elastic equations. The interaction between fluid and structure is ubiquitous in nature, arising in several areas of biological, medical and engineering sciences. Consider a doughnut-like domain: a fluid occupies the exterior sub-domain, while an elastic structure occupies the interior sub-domain. They are described by the corresponding evolution equations which present strong coupling at the interface between two domains. A key factor - a novelty over past literature - is that the structure equation includes a term defining strong damping at the interior. This affects the boundary conditions on the interface which lead to a highly unbounded ``perturbation - preventing standard methods developed for uncoupled structures to apply. Careful analysis of this effect along with the analysis of the pressure term contributed in N-S equations provides key technical - mathematical challenge. We establish several mathematical results describing the character of the overall evolution either free or else under the action of a control at the interface or at the exterior boundary
The Role of Auxiliary Stages in Gaussian Quantum Metrology
The optimization of the passive and linear networks employed in quantum metrology, the field that studies and devises quantum estimation strategies to overcome the levels of precision achievable via classical means, appears to be an essential step in certain metrological protocols achieving the ultimate Heisenberg-scaling sensitivity. This optimization is generally performed by adding degrees of freedom by means of auxiliary stages, to optimize the probe before or after the interferometric evolution, and the choice of these stages ultimately determines the possibility to achieve a quantum enhancement. In this work we review the role of the auxiliary stages and of the extra degrees of freedom in estimation schemes, achieving the ultimate Heisenberg limit, which employ a squeezed-vacuum state and homodyne detection. We see that, after the optimization for the quantum enhancement has been performed, the extra degrees of freedom have a minor impact on the precision achieved by the setup, which remains essentially unaffected for networks with a larger number of channels. These degrees of freedom can thus be employed to manipulate how the information about the structure of the network is encoded into the probe, allowing us to perform quantum-enhanced estimations of linear and non-linear functions of independent parameters
Non-adaptive Heisenberg-limited metrology with multi-channel homodyne measurements
We show a protocol achieving the ultimate Heisenberg-scaling sensitivity in the estimation of a parameter encoded in a generic linear network, without employing any auxiliary networks, and without the need of any prior information on the parameter nor on the network structure. As a result, this protocol does not require a prior coarse estimation of the parameter, nor an adaptation of the network. The scheme we analyse consists of a single-mode squeezed state and homodyne detectors in each of the M output channels of the network encoding the parameter, making it feasible for experimental applications
A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control
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.Comment: 50 pages, submitte
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