U. J. Aarsnes and O. M. Aamo, Linear stability analysis of self-excited vibrations in drilling using an infinite dimensional model, Journal of Sound and Vibration, vol.360, pp.239-259, 2016.

U. J. Aarsnes and J. S. Roman, Torsional vibrations with bit off bottom: Modeling, characterization and field data validation, Journal of Petroleum Science and Engineering, vol.163, pp.712-721, 2018.

B. Armstrong-helouvry, Stick-slip arising from stribeck friction. In Proceedings, IEEE International Conference on Robotics and Automation, pp.1377-1382, 1990.

B. Armstrong-hélouvry, P. Dupont, and C. Wit, A survey of models, analysis tools and compensation methods for the control of machines with friction, Automatica, vol.30, issue.7, pp.1083-1138, 1994.

M. Barreau, Stability analysis of coupled ordinary differential systems with a string equation -Application to a Drilling Mechanism, 2019.
URL : https://hal.archives-ouvertes.fr/tel-02194414

M. Barreau, F. Gouaisbaut, A. Seuret, and R. Sipahi, Input/output stability of a damped string equation coupled with ordinary differential system, International Journal of Robust and Nonlinear Control, vol.28, issue.18, pp.6053-6069, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01690626

M. Barreau, A. Seuret, and F. Gouaisbaut, Exponential Lyapunov stability analysis of a drilling mechanism, 57st Annual Conference on Decision and Control (CDC), pp.2952-2957, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01725416

M. Barreau, A. Seuret, F. Gouaisbaut, and L. Baudouin, Lyapunov stability analysis of a string equation coupled with an ordinary differential system, IEEE Transactions on Automatic Control, vol.63, issue.11, pp.3850-3857, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01548256

G. Bastin and J. Coron, Stability and boundary stabilization of 1-d hyperbolic systems, vol.88, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01449504

H. I. Basturk, Observer-based boundary control design for the suppression of stick-slip oscillations in drilling systems with only surface measurements, Journal of Dynamic Systems, Measurement, and Control, vol.139, issue.10, p.104501, 2017.

A. Bisoffi, M. Da-lio, A. R. Teel, and L. Zaccarian, Global asymptotic stability of a PID control system with coulomb friction, IEEE Transactions on Automatic Control, vol.63, issue.8, pp.2654-2661, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01700168

D. Bresch-pietri and M. Krstic, Output-feedback adaptive control of a wave PDE with boundary anti-damping, Automatica, vol.50, issue.5, pp.1407-1415, 2014.

C. Canudas-de-wit, F. Rubio, and M. Corchero, DOSKIL: A New Mechanism for Controlling Stick-Slip Oscillations in Oil Well Drillstrings, IEEE Transactions on Control Systems Technology, vol.16, issue.6, pp.1177-1191, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00394990

N. Challamel, Rock destruction effect on the stability of a drilling structure, Journal of Sound and Vibration, vol.233, issue.2, pp.235-254, 2000.

A. P. Christoforou and A. S. Yigit, Fully coupled vibrations of actively controlled drillstrings, Journal of sound and vibration, vol.267, issue.5, pp.1029-1045, 2003.

J. M. Coron, Control and nonlinearity. Number 136 in Mathematical Surveys and Monographs, 2007.

R. Courant and D. Hilbert, Methods of mathematical physics, 1989.

Y. Ebihara, D. Peaucelle, and D. Arzelier, S-Variable Approach to LMI-Based Robust Control, Communications and Control Engineering, vol.17, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01760625

A. F. Filippov, Classical solutions of differential equations with multivalued right-hand side, SIAM Journal on Control, vol.5, issue.4, pp.609-621, 1967.

E. Fridman, S. Mondié, and B. Saldivar, Bounds on the response of a drilling pipe model, IMA Journal of Mathematical Control and Information, vol.27, issue.4, pp.513-526, 2010.

J. K. Hale and S. M. Lunel, Effects of small delays on stability and control, Operator theory and analysis, pp.275-301, 2001.

A. Helmicki, C. A. Jacobson, and C. N. Nett, Ill-posed distributed parameter systems: A control viewpoint, IEEE Trans. on Automatic Control, vol.36, issue.9, pp.1053-1057, 1991.

D. Karnopp, Computer simulation of stick-slip friction in mechanical dynamic systems, Journal of dynamic systems, vol.107, issue.1, pp.100-103, 1985.

H. K. Khalil, Nonlinear Systems. Pearson Education, 1996.

M. Krstic, Delay compensation for nonlinear, adaptive, and PDE systems, 2009.

N. Levinson, Transformation theory of non-linear differential equations of the second order, Annals of Mathematics, pp.723-737, 1944.

X. Liu, N. Vlajic, X. Long, G. Meng, and B. Balachandran, Coupled axial-torsional dynamics in rotary drilling with state-dependent delay: stability and control, Nonlinear Dynamics, vol.78, issue.3, pp.1891-1906, 2014.

J. Löfberg, YALMIP: A toolbox for modeling and optimization in MATLAB, IEEE International Symposium on Computer Aided Control Systems Design, pp.284-289, 2005.

Ö. , On the stabilization and stability robustness against small delays of some damped wave equations, IEEE Trans. on Automatic Control, vol.40, issue.9, pp.1626-1630, 1995.

E. Navarro-lópez, An alternative characterization of bit-sticking phenomena in a multi-degree-of-freedom controlled drillstring, Nonlinear Analysis: Real World Applications, vol.10, pp.3162-3174, 2009.

E. Navarro-lópez and D. Cortes, Sliding-mode control of a multi-dof oilwell drillstring with stick-slip oscillations, Proceedings of the American Control Conference, pp.3837-3842, 2007.

E. Navarro-lópez and R. Suarez, Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings, Proceedings of the 2004 IEEE International Conference on Control Applications, vol.2, pp.1454-1460, 2004.

C. Prieur, S. Tarbouriech, and J. M. Da-silva, Wave equation with cone-bounded control laws, IEEE Trans. on Automatic Control, vol.61, issue.11, pp.3452-3463, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01448483

T. Richard, C. Germay, and E. Detournay, A simplified model to explore the root cause of stick-slip vibrations in drilling systems with drag bits, Journal of Sound and Vibration, vol.305, issue.3, pp.432-456, 2007.

C. Roman, D. Bresch-pietri, E. Cerpa, C. Prieur, and O. Sename, Backstepping observer based-control for an anti-damped boundary wave PDE in presence of in-domain viscous damping, 55th IEEE Conference on Decision and Control (CDC), 2016.
URL : https://hal.archives-ouvertes.fr/hal-01413025

M. Safi, L. Baudouin, and A. Seuret, Tractable sufficient stability conditions for a system coupling linear transport and differential equations, Systems and Control Letters, vol.110, pp.1-8, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01354073

C. Sagert, F. D. Meglio, M. Krstic, and P. Rouchon, Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling, IFAC Proceedings Volumes, vol.46, pp.779-784, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00873605

B. Saldivar, I. Boussaada, H. Mounier, and S. Niculescu, Analysis and Control of Oilwell Drilling Vibrations: A Time-Delay Systems Approach, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01123773

B. Saldivar, S. Mondié, and J. C. Vilchis, The control of drilling vibrations: A coupled PDE-ODE modeling approach, International Journal of Applied Mathematics and Computer Science, 2016.

B. Saldivar, S. Mondie, S. Niculescu, H. Mounier, and I. Boussaada, A control oriented guided tour in oilwell drilling vibration modeling, Annual Reviews in Control, vol.42, pp.100-113, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01425845

A. Serrarens, M. J. Molengraft, J. J. Kok, and L. Van-den-steen, H? control for suppressing stick-slip in oil well drillstrings, IEEE Control Systems, vol.18, pp.19-30, 1998.

A. Seuret and F. Gouaisbaut, Hierarchy of LMI conditions for the stability analysis of time-delay systems, Systems & Control Letters, vol.81, pp.1-7, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01065142

S. Tarbouriech, G. Garcia, J. M. Da-silva, and I. Queinnec, Stability and stabilization of linear systems with saturating actuators, 2011.

A. Terrand-jeanne, V. Andrieu, M. Tayakout-fayolle, and V. Martins, Regulation of inhomogeneous drilling model with a P-I controller, IEEE Transactions on Automatic Control, pp.1-1, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02083394

A. Terrand-jeanne, V. Martins, and V. Andrieu, Regulation of the downside angular velocity of a drilling string with a P-I controller, pp.2647-2652, 2018.
URL : https://hal.archives-ouvertes.fr/hal-02014813

K. Toh, M. J. Todd, and R. H. Tütüncü, SDPT3 -a MATLAB software package for semidefinite programming. Optimization methods and software, vol.11, pp.545-581, 1999.

W. R. Tucker and C. Wang, On the effective control of torsional vibrations in drilling systems, Journal of Sound and Vibration, vol.224, issue.1, pp.101-122, 1999.

M. Tucsnak and G. Weiss, Observation and control for operator semigroups, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00590673

W. Weaver, S. P. Timoshenko, and D. H. Young, Vibration problems in engineering, 1990.

, He was a Ph.D. student in LAAS-CNRS 'Laboratoire d'Analyse et d'Architecture des Systmes" in Toulouse, France under the supervision of A. Seuret and F. Gouaisbaut. He received the "Diplme de Doctorat, System Theory in 2019 from Universit Paul Sabatier, Toulouse. His research interests include stability analysis of infinite dimension systems with a particular focus on drilling systems

, He received the Diplme dIngnieur in automatic control (Engineers degree) from Ecole Centrale de Lille, France, September 1997 and the Diplme dEtudes Approfondies (Masters Degree) in the same subject from the University of Science and Technology of, 1973.

, he is an associate professor at the Paul Sabatier University (Toulouse), 2001.

A. Seuret, ) and the Masters Degree in system theory from the University of Science and Technology of Lille (France) in 2003. He received the Ph.D. degree in Automatic Control from the Ecole Centrale de Lille and the University of Science and Technology of Lille in 2006, Since 2012, he has been with the Laboratoire dArchitecture et danalyse des Systmes (LAAS), 1980.