. Dans-un-cadre-plus-mathématique and . Unprobì-eme-ouvert-proposé-dans, Problem 2] concerne la description d'un ensemble semi-algébriquè a l'aide d'un nombre minimum d'inégalités polynomiales et les liens existant avec les relaxations LMI desprobì emes d'optimisation polynomiale mentionnées

D. Arzelier, B. Clément, and D. Peaucelle, Multi-objective H2/H???/Impulse-to-Peak Control of a Space Launch Vehicle, European Journal of Control, vol.12, issue.1, pp.57-70, 2006.
DOI : 10.3166/ejc.12.57-70

B. R. Barmish, New tools for robustness of linear systems, 1994.

S. Basu, R. Pollack, and M. Roy, Algorithms in real algebraic geometry, 2005.
DOI : 10.1007/978-3-662-05355-3

URL : https://hal.archives-ouvertes.fr/hal-01083587

S. P. Bhattacharyya, H. Chapellat, and L. H. Keel, ROBUST CONTROL: THE PARAMETRIC APPROACH, 1995.
DOI : 10.1016/B978-0-08-042230-5.50016-5

V. D. Blondel, Simultaneous stabilization of linear systems, 1994.
DOI : 10.1007/3-540-19862-8

V. D. Blondel and J. N. Tsitsiklis, A survey of computational complexity results in systems and control, Automatica, vol.36, issue.9, pp.1249-1274, 2000.
DOI : 10.1016/S0005-1098(00)00050-9

S. P. Boyd, V. Balakrishnan, and P. Kabamba, A bisection method for computing the H??? norm of a transfer matrix and related problems, Mathematics of Control, Signals, and Systems, vol.28, issue.3, pp.207-219, 1989.
DOI : 10.1007/BF02551385

S. P. Boyd and C. H. Barratt, Linear controller design : limits of performance, 1991.

S. P. Boyd and L. Vandenberghe, Convex optimization, 2005.

J. V. Burke, D. Henrion, A. S. Lewis, and M. L. Overton, HIFOO - A MATLAB PACKAGE FOR FIXED-ORDER CONTROLLER DESIGN AND H??? OPTIMIZATION, Proceedings of the IFAC Symposium on Robust Control Design, 2006.
DOI : 10.3182/20060705-3-FR-2907.00059

J. V. Burke, D. Henrion, A. S. Lewis, and M. L. Overton, Stabilization via Nonsmooth, Nonconvex Optimization, IEEE Transactions on Automatic Control, vol.51, issue.11, pp.1760-1769, 2006.
DOI : 10.1109/TAC.2006.884944

A. C. Dixon, Note on the reduction of a ternary quantic to a symmetrical determinant, Proceedings of the Cambridge Philosophical Society, vol.11, pp.350-351, 1902.

J. C. Doyle, K. Glover, P. Khargonekar, and B. Francis, State-space solutions to standard H/sub 2/ and H/sub infinity / control problems, IEEE Transactions on Automatic Control, vol.34, issue.8, pp.831-846, 1989.
DOI : 10.1109/9.29425

M. Fliess, J. Lévine, P. Martin, and P. Rouchon, Sur les systèmes nonlinéaires différentiellement plats, Comptes Rendus de l'Académie des Sciences de Paris, pp.619-624, 1992.

R. M. Freund, F. Ordoñez, and K. C. Toh, Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems, Mathematical Programming, vol.95, issue.2, pp.445-475, 2007.
DOI : 10.1007/s10107-006-0035-y

F. R. Gantmacher, Theory of matrices Original version published in Russian, AMS, vol.2, 1959.

Y. Genin, Y. Hachez, Y. Nesterov, R. ¸tefan, P. Van-dooren et al., Positivity and Linear Matrix Inequalities, European Journal of Control, vol.8, issue.3, pp.275-298, 2002.
DOI : 10.3166/ejc.8.275-298

W. Gilbert, D. Henrion, J. Bernussou, and D. Boyer, Polynomial LPV synthesis applied to turbofan engines, Proceedings of the IFAC Symposium on Automatic Control in Aerospace, 2007.

K. C. Goh, M. G. Safonov, and G. P. Papavassilopoulos, Global optimization for the Biaffine Matrix Inequality problem, Journal of Global Optimization, vol.24, issue.4, pp.365-380, 1995.
DOI : 10.1007/BF01099648

D. Henrion, Stabilité des systèmes linéaires incertainsàincertainsà commande contrainte, Thèse de doctorat, Institut National des Sciences Appliquées de, 1999.

D. Henrion, D. Arzelier, and D. Peaucelle, Positive polynomial matrices and improved LMI robustness conditions, Automatica, vol.39, issue.8, pp.1479-1485, 2003.
DOI : 10.1016/S0005-1098(03)00129-8

]. D. Henrion, M. Sebek, and V. Ku?era, Positive polynomials and robust stabilization with fixed-order controllers, IEEE Transactions on Automatic Control, vol.48, issue.7, pp.1178-1186, 2003.
DOI : 10.1109/TAC.2003.814103

D. Henrion and J. B. Lasserre, GloptiPoly, ACM Transactions on Mathematical Software, vol.29, issue.2, pp.165-194, 2003.
DOI : 10.1145/779359.779363

URL : https://hal.archives-ouvertes.fr/hal-00578944

D. Henrion and J. B. Lasserre, Detecting Global Optimality and Extracting Solutions in GloptiPoly
DOI : 10.1007/10997703_15

D. Henrion and J. B. Lasserre, Convergent Relaxations of Polynomial Matrix Inequalities and Static Output Feedback, IEEE Transactions on Automatic Control, vol.51, issue.2, pp.192-202, 2006.
DOI : 10.1109/TAC.2005.863494

J. W. Helton and V. Vinnikov, Linear matrix inequality representation of sets, Communications on Pure and Applied Mathematics, vol.289, issue.5, pp.654-674, 2007.
DOI : 10.1002/cpa.20155

C. J. Herget and A. J. Laub, Special issue on computer-aided design of control systems, IEEE Control Systems Magazine, vol.2, issue.4, 1982.

N. J. Higham, Accuracy and stability of numerical algorithms, SIAM, 1996.
DOI : 10.1137/1.9780898718027

J. B. Hiriart-urruty and M. Torki, Permanently Going Back and Forth between the ``Quadratic World'' and the ``Convexity World'' in Optimization, Applied Mathematics and Optimization, vol.45, issue.2, pp.169-184, 2002.
DOI : 10.1007/s00245-001-0034-6

J. B. Hiriart-urruty, Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization, SIAM Review, vol.49, issue.2, pp.255-273, 2007.
DOI : 10.1137/050633500

URL : https://hal.archives-ouvertes.fr/hal-00635746

T. Kailath, Linear systems, 1980.

F. Kahl and D. Henrion, Globally Optimal Estimates for Geometric Reconstruction Problems, International Journal of Computer Vision, vol.20, issue.7, pp.3-15, 2007.
DOI : 10.1007/s11263-006-0015-y

D. Knittel, M. Vedrines, D. Henrion, and P. Pagilla, Robust H?? Fixed Order Control Strategies for Large Scale Web Winding Systems, IEEE International Symposium on Intelligent Control, 2006.
DOI : 10.1109/ISIC.2006.285637

D. Knittel, M. Vedrines, D. Henrion, and P. Pagilla, Fixed-Order H<sub>&#x0221E;</sub> Decentralized Control with Model Based Feedforward for Elastic Web Winding Systems, Conference Record of the 2006 IEEE Industry Applications Conference Forty-First IAS Annual Meeting, 2006.
DOI : 10.1109/IAS.2006.256701

M. Ko?vara and M. Stingl, PENNON : a code for convex nonlinear and semidefinite programming, Optimization Methods and Software, vol.8, issue.3, pp.317-333, 2003.

M. Ko?vara, F. Leibfritz, M. Stingl, and D. Henrion, A nonlinear SDP algorithm for static output feedback problems in COMPLIB, Proceedings of the IFAC World Congress on Automatic Control, 2005.

V. Ku?era, Discrete linear control -the polynomial equation approach, 1979.

V. Ku?era, Analysis and design of discrete linear control systems, 1991.

H. Kwakernaak, Robust control and H???-optimization???Tutorial paper, Automatica, vol.29, issue.2, pp.255-273, 1993.
DOI : 10.1016/0005-1098(93)90122-A

Y. Labit, D. Peaucelle, and D. Henrion, SEDUMI INTERFACE 1.02: a tool for solving LMI problems with SEDUMI, Proceedings. IEEE International Symposium on Computer Aided Control System Design, 2002.
DOI : 10.1109/CACSD.2002.1036966

J. B. Lasserre, Global Optimization with Polynomials and the Problem of Moments, SIAM Journal on Optimization, vol.11, issue.3, pp.796-817, 2001.
DOI : 10.1137/S1052623400366802

J. B. Lasserre, C. Prieur, and D. Henrion, Nonlinear optimal control: approximations via moments and LMI-relaxations, Proceedings of the 44th IEEE Conference on Decision and Control, 2005.
DOI : 10.1109/CDC.2005.1582395

J. Löfberg, YALMIP : a toolbox for modeling and optimization in MATLAB, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508), 2004.
DOI : 10.1109/CACSD.2004.1393890

Y. Nesterov and A. Nemirovskii, Interior point polynomial algorithms in convex programming, SIAM, 1994.

P. C. Parks and V. Hahn, Stability theory Original version published in German, 1981.

P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming, vol.96, issue.2, pp.293-320, 2003.
DOI : 10.1007/s10107-003-0387-5

D. Peaucelle and D. Arzelier, Robust multi-objective control toolbox, Proceedings of the IEEE Symposium on Computer-Aided Control System Design, 2006.

J. W. Polderman and J. C. Willems, Introduction to mathematical systems theory : a behavorial approach, 1998.
DOI : 10.1007/978-1-4757-2953-5

B. T. Polyak, Convexity of Quadratic Transformations and Its Use in Control and Optimization, Journal of Optimization Theory and Applications, vol.6, issue.3, pp.552-583, 1998.
DOI : 10.1023/A:1021798932766

L. Polyx, The Polynomial Toolbox for Matlab. Version 2.5, 1998.

S. Prajna, A. Papachristodoulou, and P. A. Parrilo, Introducing SOS- TOOLS : a general purpose sum of squares programming solver, Proceedings of IEEE Conference on Decision and Control, 2002.

S. Prajna, P. A. Parrilo, and A. Rantzer, Nonlinear Control Synthesis by Convex Optimization, IEEE Transactions on Automatic Control, vol.49, issue.2, pp.310-314, 2004.
DOI : 10.1109/TAC.2003.823000

L. Reberga, Commande robuste multivariable des turboréacteurs

L. Reberga, D. Henrion, J. Bernussou, and F. Vary, LPV MODELING OF A TURBOFAN ENGINE, Proceedings of the IFAC World Congress on Automatic Control, 2005.
DOI : 10.3182/20050703-6-CZ-1902.00488

R. T. Rockafellar, Lagrange Multipliers and Optimality, SIAM Review, vol.35, issue.2, pp.183-238, 1993.
DOI : 10.1137/1035044

H. H. Rosenbrock, State-Space and Multivariable Theory, IEEE Transactions on Systems, Man, and Cybernetics, vol.2, issue.2, 1970.
DOI : 10.1109/TSMC.1972.4309115

C. W. Scherer, LMI Relaxations in Robust Control, European Journal of Control, vol.12, issue.1, pp.3-29, 2006.
DOI : 10.3166/ejc.12.3-29

S. Skogestad and I. Postlethwaite, Multivariable feedback control : analysis and design, 1996.

L. N. Trefethen and M. Embree, Spectra and Pseudospectra, 2005.
DOI : 10.1007/978-3-662-03972-4_6

V. Vinnikov, Self-adjoint determinantal representations of real plane curves, Mathematische Annalen, vol.28, issue.1, pp.453-479, 1993.
DOI : 10.1007/BF01445115

K. Zhou, J. C. Doyle, and K. Glover, Robust and optimal control, 1996.

J. C. Zúñiga-anaya, Algorithmes numériques pour les matrices polynomiales avec applications en commande, Thèse de doctorat, Institut National des Sciences Appliquées de, 2005.