The realization of a canonical system with dissipative boundary conditions at one end of a segment in terms of the coefficient of dynamic flexibility

1976 ◽  
Vol 16 (3) ◽  
pp. 335-352
Author(s):  
D. Z. Arov
Keyword(s):  
Boundary Conditions ◽  
Canonical System ◽  
Dissipative Boundary

scholarly journals Eigenvalues for Maxwell’s equations with dissipative boundary conditions

2016 ◽  
Vol 99 (1-2) ◽  
pp. 105-124 ◽  
Author(s):  
Ferruccio Colombini ◽  
Vesselin Petkov ◽  
Jeffrey Rauch
Keyword(s):  
Boundary Conditions ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Dissipative Boundary

scholarly journals Asymptotic Properties of Linear Field Equations in Anti-de Sitter Space

2019 ◽  
Vol 374 (2) ◽  
pp. 1125-1178 ◽  
Author(s):  
Gustav Holzegel ◽  
Jonathan Luk ◽  
Jacques Smulevici ◽  
Claude Warnick
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Properties ◽  
Uniform Boundedness ◽  
Global Dynamics ◽  
Decay Estimates ◽  
De Sitter ◽  
Field Equations ◽  
Sitter Spacetime ◽  
Loss Of Derivatives ◽  
Dissipative Boundary

Abstract We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

scholarly journals Uniform exponential stabilisation of serially connected inhomogeneous Euler-Bernoulli beams

2020 ◽  
Vol 26 ◽  
pp. 110
Author(s):  
Björn Augner
Keyword(s):  
Boundary Conditions ◽  
Modulus Of Elasticity ◽  
Spatial Dependence ◽  
Mass Density ◽  
Coupled System ◽  
The Other ◽  
Exponential Energy Decay ◽  
A Chain ◽  
Dissipative Boundary ◽  
Dependent Mass

We consider a chain of Euler-Bernoulli beams with spatial dependent mass density, modulus of elasticity and area moment which are interconnected in dissipative or conservative ways and prove uniform exponential energy decay of the coupled system for suitable dissipative boundary conditions at one end and suitable conservative boundary conditions at the other end. We thereby generalise some results of G. Chen, M.C. Delfour, A.M. Krall and G. Payre from the 1980’s to the case of spatial dependence of the parameters.

scholarly journals Stability of a flexible structure with destabilizing boundary conditions

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160109 ◽  
Author(s):  
M. Shubov ◽  
V. Shubov
Keyword(s):  
Boundary Conditions ◽  
Control Parameter ◽  
The Other ◽  
Mode Shapes ◽  
Beam Model ◽  
Control Parameters ◽  
The Stability ◽  
Dissipative Boundary ◽  
The Right ◽  
Observation Vector

The Euler–Bernoulli beam model with non-dissipative boundary conditions of feedback control type is investigated. Components of the two-dimensional input vector are shear and moment at the right end, and components of the observation vector are time derivatives of displacement and slope at the right end. The codiagonal matrix depending on two control parameters relates input and observation. The paper contains five results. First, asymptotic approximation for eigenmodes is derived. Second, ‘the main identity’ is established. It provides a relation between mode shapes of two systems: one with non-zero control parameters and the other one with zero control parameters. Third, when one control parameter is positive and the other one is zero, ‘the main identity’ yields stability of all eigenmodes (though the system is non-dissipative). Fourth, the stability of eigenmodes is extended to the case when one control parameter is positive, and the other one is sufficiently small. Finally, existence and properties of ‘deadbeat’ modes are investigated.

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