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. Muñoz Rivera LNCC IM-UFRJ - PowerPoint PPT Presentation

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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Presentation Transcript

Our problem:

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

Polynomialstability

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

This is a join work with Luci Fatori:

Paraná – Brasil

e-mail: lucifatori@uel.br

Our interest is to prove that the sufficient condition of Liu

and Rao is also a necessary condition.

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.

The Infenitesimal generator of the semigroup is

We denote the associated semigroup as

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.

and Differentiable when

Our

Contribution

Our contribution is about polynomial stability for

Our stability result to damped wave equation is

Our

contribution

to Bresse system