Empathy theory and the Laplace transform

Sauer, Niko

doi:10.4064/-38-1-325-338

Cited by 17 publications

(24 citation statements)

References 10 publications

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“…The evolution from an initial state in Y to a solution in the space X is generated by the operator pair −A B → Y × Y , which is called the generating pair of the double family S t Y → X t > 0 , E t Y → Y t > 0 whenever S t y solves Pr AEP . The characterization of generators of such double families of evolution operators may be found in [30,31]. It is evident that the semigroup E t is determined by the operators A and B in a unique way and that there exists a unique bounded linear operator C Y → X such that S t = CE t .…”

Section: Pr (Aep)mentioning

confidence: 99%

“…In this appendix, we list definitions and results from the theory of double families of evolution operators (empathy theory), as developed by Sauer in [29][30][31], which are important in this study. We recall (Section 1.3) that the concept of double families of evolution operators was developed with the aim of treating implicit evolution equations of the form Find U such that…”

Section: Appendixmentioning

confidence: 99%

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Thermal Effects in a Two-Dimensional Hybrid Elastic Structure

Dalsen

2002

Journal of Mathematical Analysis and Applications

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In this paper we consider a two-dimensional hybrid thermo-elastic structure consisting of a thermo-elastic plate which has a beam attached to its free end. We show that the initial-boundary-value problem for the interactive system of partial differential equations which take account of the mechanical strains/stresses and the thermal stresses in the plate and the beam, can be associated with a uniformly bounded evolution operator. It turns out that the interplay of parabolic dynamics due to the thermal effects in the hybrid structure and the hyperbolic dynamics associated with the elasticity of the structure yields analyticity for the entire system. This result yields solvability for the problem under optimal initial freedom of the displacement, velocity, and temperature in the plate and the beam, while uniform stability is readily available.  2002 Elsevier Science

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Section: Pr (Aep)mentioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

Thermal Effects in a Two-Dimensional Hybrid Elastic Structure

Dalsen

2002

Journal of Mathematical Analysis and Applications

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“…It has been shown [11] that for the study of implicit evolution equations, 'double families' of evolution operators play a role similar to that of semigroups. In this approach cause and effect are described in different Banach spaces Y (cause) and X (effect).…”

Section: Introductionmentioning

confidence: 99%

“…For partial differential equations with dynamic boundary conditions it may easily happen that B is not closable [16]. In order to cover a wider range of situations a dynamic systems approach to implicit evolution equations was developed [10,11]. The theory is based on double families of evolution operators and the empathy relation mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Double families of integrated evolution operators

Brown

Sauer

2004

J.evol.equ.

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A theory of double families of evolution operators, which involves mappings between different spaces, is presented. Causality is in a form analogous to that in integrated semi-groups, but involves an integrated relationship between members of both families. The approach is from a dynamic systems point of view, and focuses very much on the trajectories defined by the evolution operators and their relationship to the solutions of implicit evolution equations. Illustrative examples concerned with dynamic boundary conditions are provided.

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“…Finally we note that, as in our study of a non-linear thermo-elastic Euler-Bernoulli beam-rigid body structure with rotational inertia, an analytic evolution operator is here, in the absence of thermal e ects in the structure, not attainable. Hence the theory of double families of evolution operators developed by Sauer [25], which has particular advantages in the analytic case (see e.g. Reference [18]), will not be employed in this study.…”

Section: Riessner-mindlin Plate-timoshenko Beam Hybrid Structurementioning

confidence: 99%

Dynamic boundary stabilization of a Reissner–Mindlin plate with Timoshenko beam

Dalsen

2004

Math Methods in App Sciences

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SUMMARYThis paper is concerned with well-posedness results for a mathematical model for the transversal vibrations of a two-dimensional hybrid elastic structure consisting of a rectangular Reissner-Mindlin plate with a Timoshenko beam attached to its free edge. The model incorporates linear dynamic feedback controls along the interface between the plate and the beam. Classical semigroup methods are employed to show the unique solvability of the coupled initial-boundary-value problem. We also show that the energy associated with the system exhibits the property of strong stability.

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Empathy theory and the Laplace transform

Cited by 17 publications

References 10 publications

Thermal Effects in a Two-Dimensional Hybrid Elastic Structure

Thermal Effects in a Two-Dimensional Hybrid Elastic Structure

Double families of integrated evolution operators

Dynamic boundary stabilization of a Reissner–Mindlin plate with Timoshenko beam

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