W. Est-bien-une-fonction-différentiable and M. Propre, La dérivée de W Ie long des trajectoires du système bouclé (5.1X5.10) est donnée par W(*,r1 : Vçs.et(r) + VV(r)G(ae,a)y -llallr*k(*,s)

C. Lvv, < ll Vtz(r)c( r, y)ll^llyll*, il vient que w(r,a) S Vç.ut(r) + llvr( r)G(ae,a)

. La-partie-droite-de-i-'inégalité, 13) peut être considérée comme un polynôme de degré deux en llyll-avec le discriminant suivant L(*, a) : | | V lz(r ) G (r, ùll'* + +t (*, a)Vts.q(r)

L. En-utilisant and . Inégalités, 8) et (5.9), on obtient A(r,g) <0 pour ll"ll">M, finalement W(*,y)

. Dans-cette-annexe, nous reprenons, avec de légères modifications, un lemme proposé dans

. Preuve, En vertu du théorème inverse de Lyapunov, il existe une fonction I

. Montrer, 9) revient donc à prouver que 9À(y) ) 0 pour tout

. Il-est-clair-que-pour, 0, go(a) : f (0,y) est strictement positive pour tort y I 0

. [. Bibliographie and . Aeyels, Stabilization of a class of nonlinear systems by a smooth feedback control, Systems ?j Control Letters, vol.5, pp.289-294, 1985.

. A. Abs, A. Andreini, G. Bacciotti, and . Stefani, Global stabilizability of homogeneous vector fields of odd degree, Systems ?i Control Letters, vol.10, pp.2-2

Z. Artstein, Stabilization with relaxed controls, Nonlinear Analysis: Theory, Methods & Applications, vol.7, issue.11, pp.1163-1173, 1983.
DOI : 10.1016/0362-546X(83)90049-4

]. A. Ba and . Bacciotti, Local stabilization of nonlinear control systems, Series on aduances in mathematics for applied sciences, 1992.

. M. Bm-]-w, R. Boothby, and . Marino, Feedback linearization of planar noniinear systems II, Proc. 28th IEEE-CDC Conference), pp.1970-1974, 1999.

]. R. Br and . Brockett, Differential Geometric Control Theory, chapter Asymptotic stability and feedback stabilization, pp.181-182, 1983.

. Bi-]-c, A. Byrnes, and . Isidori, New results and examples in nonlinear feedback stabilization, Systems ?i Control Letters, vol.12, pp.437-442

. I. Biw-]-c, A. Byrnes, J. C. Isidori, and . Willems, Passivity, feedback equivalence and the global stabilization of minimum phase nonlinear systems, IEEE transaction on control,nol as, vol.36, issue.11, pp.228-240, 1991.

G. [. Chabour, J. C. Sallet, and . Vivalda, Stabilization of nonlinear systems: A bilinear approach, Mathematics of Control, Signals, and Systems, vol.2, issue.3, pp.224-246
DOI : 10.1007/BF01211621

]. C. Col and . Coleman, Asymptotic stability in 3-space, Contributions to the theory of Nonlinear Oscillations, V; Annals of Mathernathi,cs Studies. 45, 1960.

]. W. Cop, A survey of quadratic systems, J. Diff, vol.82, pp.293-304, 1966.

]. J. Col and . Coron, A necessary condition for feedback stibilisation, Syst. Contr. Lett, vol.14, pp.227-232

J. M. Coron, Linearized Control Systems and Applications to Smooth Stabilization, SIAM Journal on Control and Optimization, vol.32, issue.2, pp.358-386, 1994.
DOI : 10.1137/S0363012992226867

J. M. Coron, On the Stabilization in Finite Time of Locally Controllable Systems by Means of Continuous Time-Varying Feedback Law, SIAM Journal on Control and Optimization, vol.33, issue.3, pp.805-833
DOI : 10.1137/S0363012992240497

E. [. Coron and . Kerai, Explicit feedback stabilizing the attitude spacecraft with tow control torques. à paraitre dans Automatica, Coron and L. Praly. Adding an integrator for the stabilization Systerns ?i Control Letters, pp.89-104, 1991.

]. T. Da and . Date, Classification and Analysis of Two-Dimensional Real Homogeneous Quadratic Differential Equation, J . Diff. Eq, vol.32, pp.311-344, 1979.

[. Date and M. Iri, Canonical forms of real homogeneous quadratic transformations, Journal of Mathematical Analysis and Applications, vol.56, issue.3, pp.650-682, 1976.
DOI : 10.1016/0022-247X(76)90031-7

]. W. Dml, C. F. Dayawansa, and . Martin, Two examples of stabilizable second order systems, Proc. Montana Conference on Computation and Control, 1988.

C. [. Dayawansa and . Martin, A remark on a theorem of Andreini, Bacciotti and Stefan\. Systems ?i Control Letters, pp.363-364, 1989.

. P. Dmk-]-w, C. F. Dayawansa, G. Martin, and . Knowles, Asymptotic stabilization of a class of smooth two-dimensional systems, SIAM J. Control and Optimization, vol.23, pp.1321-1349, 1990.

. P. Dms-]-w, C. F. Dayawansa, S. Martin, and . Samelson, Asymptotic stabilization of a generic class of three dimensional homogeneous quadratic systems, Systems I Control Letters, vol.24, pp.115-123, 1995.

. J. Dp-]-r, L. M. Dickson, and . Perker, Bounded Quadratic System in the plane, J. Difl:. Eq, vol.7, pp.251-273, 1970.

]. M. Bibliographie-[-gt and . Gromov, Partial Differential Relations, Ergebnisse Mathernatik7, F.Olge g, 1986.

]. X. Hu and . Hu, Stabilization of planar nonlinear systems by polynomial feedback control. Systems ?i Control Letters, pp.177-185, 1994.

]. H. He1 and . Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. Differential Equations, Stability and control, S.ELAYDI ed., Lecture Notes in Pure and AppI, Math, vol.27, pp.249-260, 1990.

]. H. He2 and . Hermes, Nilpotent and high-order approximations of vector field systems, SIAM Reaiew, vol.33, issue.2, pp.238-264, 1991.

]. H. He3 and . Hermes, Asymptotic Stabilizing Feedback Controls Journal of Differential Equati,ons,92, pp.76-89, 1991.

]. H. He and . Hermes, Asymptotic stabilization of planar systems, Systems ?j Control Letters, vol.17, pp.437-443

]. H. He5 and . Hermes, Homogeneous feedback controls for homogeneous systems, Systerns ?j Control Letters, vol.24, pp.7-11, 1995.

. A. Ijo, H. Iggidr, R. Jghima, and . Outbib, Global Feedback Stabilization of Homogeneous Polynomial Systems

. A. Is, G. Iggidr, and . Sallet, Nonlinear stabilization by adding an integrators, I{Y- BELNETII, vol.30, issue.4 5, pp.499-506, 1994.

. A. Iv, J. Iggidr, and . Vivalda, Global stabilization of homogeneous polynomial systems in IR3. Nonlinear Analysis Theory, Methods ?l Apptications,lS, pp.1181-1186, 1992.

]. A. Ir and . Isidori, Nonlinear Control Systems, 1989.

]. H. Uol, R. Jghima, and . Outbib, Lagrange stability of a class of quadratic systems in lR3, Présenté à la deuxième conférence intérnationale sur les équations différentielles. Marrakech, juin 95

H. Jghima and R. Outbib, A remark on the stabilization of homogeneous polynomial systems, Applied Mathematics Letters, vol.9, issue.2, pp.47-48, 1996.
DOI : 10.1016/0893-9659(96)00010-9

H. Jghima and R. Outbib, Comments on the stabilization of nonlinear systems by adding an integrator, IEEE transaction on control. À paraitre

H. Jghima and R. Outbib, Stabilité au sens de Lagrange d'une classe de systèmes. quadratiques dans n3, C. R. Acad. Sci. Paris, t, vol.322, pp.935-938, 1996.

J. [. Jurdjevic and . Quinn, Controllability and stability, Journal of Differential Equations, vol.28, issue.3, pp.381-389, 1978.
DOI : 10.1016/0022-0396(78)90135-3

]. B. Kalitine, Sur la stabilité des ensembles compacts positivement invariants des systèmes dynamiques, R.A.I.R.O. Automatique/ Systems Analysis and Control, vol.16, pp.3275-286, 1982.

M. Kâwski, Homogeneous stabilizing feedback laws. Control-Theory and Adaanced Technology, pp.497-516, 1990.

M. Kawski, Stabilization of nonlinear systems in the plane. Systems I Control Letters, pp.169-175, 1990.

]. D. Ko and . Kodistscheck, Adaptive techniques for mechanical systems, Fifth Yale workshop on adaptiae systems, p.259, 1987.

. V. Ks-]-p, H. J. Kokotovic, and . Sussmann, A positive real condition for global stabilization of nonlinear systems, Systems ?i Control Letters, vol.13, pp.125-133, 1989.

. J. Ll, S. Lasalle, and . Lefschetz, Stability by Liapunoa's direct method with applications, 1961.

]. L. Mar and . Markus, Quadratic differential equations and non-associative algebras, Ann. Mâth. Studi,es, pp.185-213, 1960.

[. Markus and H. Yamabe, Global stability criteria for differential systems, Oseka Math. J, vol.12, pp.305-317, 1960.

. [. Massera, Contributions to Stability Theory, The Annals of Mathematics, vol.64, issue.1, pp.182-206, 1956.
DOI : 10.2307/1969955

]. R. Oul and . Outbib, Stabilisation d'une classe de systèmes affines en contrôles, Proceedings of European control conference, pp.480-484, 1991.

]. R. Bibliographieou2 and . Outbib, Sur Ia stabilisation globale des systèmes non linéaires par retour d'état régulier, Thèse de l'uni

G. [. Outbib and . Sallet, Stabilizability of the angular velocity of a rigid body revisited, Systems & Control Letters, vol.18, issue.2, pp.93-98, 1992.
DOI : 10.1016/0167-6911(92)90013-I

R. Outbib and G. Sallet, Stabilization by dynamic feedback. Workshop on Nonsmooth Analysis and Differential Geometric Methods in Optimal Control. I. M. A Minneapolis, 1993.

]. L. Ro and . Rosier, Etude de quelques problemes de stabilisation. Thèse de I'uniaérsité de Paris XI, 1993.

]. C. Sa and . Samson, Velocity and torque feedback control of nonholonomic cart Aduanced , Robot, Control, Proceeding's of the International Workshop on Nonlineair and Adaptativ Control: Issuses In Robotics, Lecture Note in Control and Information Sciences, vol.162, pp.125-151, 1990.

. A. Sks, P. V. Saberi, H. J. Kokotovic, and . Sussmann, Global stabilization of partially linear composed systems, SIAM J. Control and Optirnisation, vol.23, pp.1491-1503, 1990.

R. [. Seibert and . Suarez, Global stabilization of nonlinear cascade systems, Systems & Control Letters, vol.14, issue.4, pp.347-352, 1990.
DOI : 10.1016/0167-6911(90)90056-Z

R. [. Seibert and . Suarez, Global stabilization of a certain class of nonlinear systems, Systems & Control Letters, vol.16, issue.1, pp.17-23, 1991.
DOI : 10.1016/0167-6911(91)90024-9

]. E. So1 and . Sontag, Smooth stabilization implies coprime factorization, IEEE transaction on control N, vol.34, pp.435-443, 1989.

]. E. So2 and . Sontag, Mathematical Control Theory Deterministic finite dimentional Systems, Texts in applied Mathematics, vol.6, 1990.

]. E. Sosu, H. J. Sontag, and . Sussmann, Further comments on the stabilizability of the angular velocity of a rigid body. Systems and, Control Letters, pp.213-217, 1988.

]. R. Sp and . Sepulcheres, Contributions to nonlinear control systems analysis by means of the direct method of Liapunov

]. H. Suk, P. V. Sussmann, and . Kokotovic, The Peaking Phenomenom and the Global Stabilization of Nonlinear Systems, IEEE transaction on control,aol as, vol.36, issue.4, pp.424-440, 1991.