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## Stability and instability of compactly coupled wave equations with delay terms in the feedbacks

Time delay exists in many practical systems such as engineering systems (see AbdelRohman [2], [3], Agrawal and Yang [4],[5], Phohomsiri et al [35]), biological systems (Batzel et al [6]), etc… It may be a source of instability. In fact, it is by now well known that certain innite-dimensional second order systems are destabilized by small time delay in the damping (Datko et al [9], Datko [8]). On the other hand, it may have a stabilizing eect and it could improve the system performances, (see Abdallah et al [1], Chiasson and Abdallah [7], Niculescu et al [31], Kwon et al [14],… ).

Thus, the stability analysis of time delay system is an important subject for investigations from both theoretical and practical point of view. The stability analysis of control systems governed by ordinary dierential equations with constant or time-varying delays has been studied by many researchers (Kolmanovskii and Myshkis [13], Niculescu [30], Richard [37], Gu et al [11],… ). Two methods, one is based on Lyapunov-Razumikhin functional and the other is based on Lyapunov-Krasovskii functional , are widely used in order to nd a delay independent or a delay dependent stability conditions.

Stability of partial dierential equations with delay has also attracted the attention of many authors. Datko et al [9] analyzed the eect of time delay in boundary feedback stabilization of the one-dimensional wave equation. They showed that an almost arbitrary small time delay destabilize the system which is exponentially stable in the absence of delay. In Datko [8], the author presented two examples of hyperbolic partial dierential equations which are stabilized by boundary feedback controls and then destabilized by small time delays in these controls. Li and Liu [20] proved that stabilization of parabolic systems is robust with respect to small time delays in their feedbacks, however stabilization of innite-dimensional conservative systems is not. Xu et al [39] established sucient conditions ensuring the stability of one dimensional wave equation with a constant time delay term in the boundary feedback controller using spectral methods. More precisely, they split the controller into two parts: one has no delay and the other has a time delay. They showed that if the constant gain of the delayed damping term is smaller (larger) than the undelayed one then the system is exponential stable (unstable). When the two constant gains are equal, they proved that the system is asymptotically stable for some time delays. This result have been extended to the multidimensional wave equation with a delay term in the boundary or internal feedbacks by Nicaise and Pignotti [28]. Similarly to (Xu et al [39]), they established an exponential stability result in the case where the constant gain of the delayed term is smaller than the undelayed one. This result is obtained by introducing an appropriate energy function and by using a suitable observability estimate. In the other cases, they constructed a sequence of time delays for which instability occurs. Nicaise and Rebiai [29] considered the multidimensional Schrödinger equation with a delay term in the boundary or internal feedbacks. Adopting the approach of (Nicaise and Pignotti [28]), they established stability and instability results.

### Stability of coupled Euler-Bernoulli equations with delay terms in the internal feedbacks

In this thesis we have studied stability problems for some systems governed by partial dierential equations:

• Coupled wave equations.

• Transmission wave equation.

• Coupled Euler-Bernoulli equations.

with time delays in the boundary or internal feedbacks.

The approach we adopted uses:

• An appropriate energy function.

• Observability estimate type for the corresponding homogeneous system whose proof combines either classical or Carleman multiplier techniques and compactnessuniqueness argument.

There are several extensions of the results obtained in this thesis. For example the following questions can be considered for future work :

• Stability of coupled wave or Euler-Bernoulli equations with delay term in one of the boundary feedback without assuming that the constant gain of the delayed term is less than of the undelayed one.

• Stability of coupled wave or Euler-Bernoulli equations with time delays in the non linear (boundary or internal) feedbacks.

• Stabilization of wave or Euler-Bernoulli system with time delays in the boundary feedback by an internal feedback.

**Abstract **

**Acknowledgements **

**Introduction**

**Chapter 1. Stability and instability of compactly coupled wave equations with delay terms in the feedbacks**

1.1. Stability of compactly coupled wave equations with delay terms in the boundary feedbacks

1.2. Stability of compactly coupled wave equations with delay terms in the internal feedbacks

1.3. Instability

**Chapter 2. Stability of the transmission wave equation with a delay term in the boundary feedback **

2.1. Introduction

2.2. Main result

2.3. Well-posedness

2.4. Proof of the main result

**Chapter 3. Stability of the transmission wave equation with delay terms in the boundary and internal feedbacks**

3.1. Introduction

3.2. Main result

3.3. Well-posedness

3.4. Proof of the main result

**Chapter 4. Stability of coupled Euler-Bernoulli equations with delay terms in the boundary feedbacks**

4.1. Introduction

4.2. Main result

4.3. Well-posedness

4.4. Proof of the main result

**Chapter 5. Stability of coupled Euler-Bernoulli equations with delay terms in the internal feedbacks**

5.1. Introduction

5.2. Main result

5.3. Well-posedness

5.4. Proof of the main result

**Conclusion **

**Appendix A**

**Appendix B**

**Bibliography**