# Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks

## Abstract

In this paper, we study the energy decay rate for the elastic Bresse system in one-dimensional bounded domain. The physical system consists of three wave equations. The two wave equations about the rotation angle and the longitudinal displacement are damped by two locally distributed feedbacks at the neighborhood of the boundary. Then indirect damping is applied to the equation for the transverse displacement of the beam through the coupling terms. We will establish the exponential stability for this system in the case of the same speed of propagation in the equation for the vertical displacement and the equation for the rotation angle of the system. When the wave speeds are different, nonexponential decay rate is proved and a polynomial-type decay rate is obtained. The frequency domain method and the multiplier technique are applied.

- Authors:

- Universite Libanaise, Faculte Des Sciences 1, Hadath, Beyrouth (Lebanon)
- Universite Libanaise, Faculte Des Sciences 4, Nabatieh (Lebanon)

- Publication Date:

- OSTI Identifier:
- 21476588

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 51; Journal Issue: 10; Other Information: DOI: 10.1063/1.3486094; (c) 2010 American Institute of Physics; Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; COUPLING; DAMPING; ONE-DIMENSIONAL CALCULATIONS; POLYNOMIALS; ROTATION; STABILITY; VELOCITY; WAVE EQUATIONS; DIFFERENTIAL EQUATIONS; EQUATIONS; FUNCTIONS; MATHEMATICAL SOLUTIONS; MOTION; PARTIAL DIFFERENTIAL EQUATIONS

### Citation Formats

```
Wehbe, Ali, and Youssef, Wael.
```*Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks*. United States: N. p., 2010.
Web. doi:10.1063/1.3486094.

```
Wehbe, Ali, & Youssef, Wael.
```*Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks*. United States. doi:10.1063/1.3486094.

```
Wehbe, Ali, and Youssef, Wael. Fri .
"Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks". United States. doi:10.1063/1.3486094.
```

```
@article{osti_21476588,
```

title = {Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks},

author = {Wehbe, Ali and Youssef, Wael},

abstractNote = {In this paper, we study the energy decay rate for the elastic Bresse system in one-dimensional bounded domain. The physical system consists of three wave equations. The two wave equations about the rotation angle and the longitudinal displacement are damped by two locally distributed feedbacks at the neighborhood of the boundary. Then indirect damping is applied to the equation for the transverse displacement of the beam through the coupling terms. We will establish the exponential stability for this system in the case of the same speed of propagation in the equation for the vertical displacement and the equation for the rotation angle of the system. When the wave speeds are different, nonexponential decay rate is proved and a polynomial-type decay rate is obtained. The frequency domain method and the multiplier technique are applied.},

doi = {10.1063/1.3486094},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 10,

volume = 51,

place = {United States},

year = {2010},

month = {10}

}