Optimality on Polynomial decay of semigroups in Elasticity
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Optimality on Polynomial decay of semigroups in Elasticity By Jaime E. Muñoz Rivera LNCC IM-UFRJ. Our problem :. Given a semigroup of contractions T(t) defined over Hilbert space H, find the best number p such that. know result about Polynomial stability.

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Optimality on Polynomial decay of semigroups in Elasticity By Jaime E. Muñoz Rivera LNCC IM-UFRJ

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Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Optimality on Polynomial decay of semigroups in Elasticity

By

Jaime E. Muñoz Rivera

LNCC

IM-UFRJ


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Our problem:

Given a semigroup of contractions T(t) defined over Hilbert space H, find the best number p such that


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

knowresultabout

Polynomialstability


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Liu and Rao proved a sufficient condition to get the polynomial decay of a semigroup


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Removing logarits, Liu and Rao’s Theorem can be written as:


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

A sufficient and a necessary condition to polynomial decay was given by J. Pruss 2006


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Other importante result due to Pruss 2006


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Pruss method give a necessary and a sufficient condition to prove polynomial estability

The problem is that it is not a simple task to estimate fractional powers of the operator of I.G.S.

It is more easy to deal with the sufficient condition given by Liu and Rao.

Our purporse is to show that the sufficient condition of Liu and Rao is also a necessary condition


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

The nexus between Liu-Rao and Pruss Characterization is given by a result due to


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

This is a join work with Luci Fatori:

Estadual University of Londrina

Paraná – Brasil

e-mail: [email protected]

Our interest is to prove that the sufficient condition of Liu

and Rao is also a necessary condition.


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Our main result is the following necessary condition

This result will be important to show when a rate of decay is optimal.

The proof is based on Pruss necessary condition and Latushkin –Shvidkoy result.


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Aplications


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

The Infenitesimal generator of the semigroup is

We denote the associated semigroup as


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Previousresult


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

That system was studied by Chen and Triggiani. They proved that the semigroup is analytic if

The authors solved the conjetures of G. Chen and D. L. Russel on structural damping for elastic systems.


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Z. Liu and K. Liu, proved that the semigroup is analytic when

and Differentiable when


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Our

Contribution


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Our contribution is about polynomial stability for

Our stability result to damped wave equation is


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Idea oftheproofoftheoptimality:


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Our

contribution

to Bresse system


Optimality on polynomial decay of semigroups in elasticity by jaime e mu oz rivera lncc im ufrj

Thanks


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