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) ,
< ll Vtz(r)c( r, y)ll^llyll*, il vient que w(r,a) S Vç.ut(r) + llvr( r)G(ae,a) ,
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) ,
8) et (5.9), on obtient A(r,g) <0 pour ll"ll">M, finalement W(*,y) ,
nous reprenons, avec de légères modifications, un lemme proposé dans ,
En vertu du théorème inverse de Lyapunov, il existe une fonction I ,
9) revient donc à prouver que 9À(y) ) 0 pour tout ,
0, go(a) : f (0,y) est strictement positive pour tort y I 0 ,
Stabilization of a class of nonlinear systems by a smooth feedback control, Systems ?j Control Letters, vol.5, pp.289-294, 1985. ,
Global stabilizability of homogeneous vector fields of odd degree, Systems ?i Control Letters, vol.10, pp.2-2 ,
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
Local stabilization of nonlinear control systems, Series on aduances in mathematics for applied sciences, 1992. ,
Feedback linearization of planar noniinear systems II, Proc. 28th IEEE-CDC Conference), pp.1970-1974, 1999. ,
Differential Geometric Control Theory, chapter Asymptotic stability and feedback stabilization, pp.181-182, 1983. ,
New results and examples in nonlinear feedback stabilization, Systems ?i Control Letters, vol.12, pp.437-442 ,
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. ,
Stabilization of nonlinear systems: A bilinear approach, Mathematics of Control, Signals, and Systems, vol.2, issue.3, pp.224-246 ,
DOI : 10.1007/BF01211621
Asymptotic stability in 3-space, Contributions to the theory of Nonlinear Oscillations, V; Annals of Mathernathi,cs Studies. 45, 1960. ,
A survey of quadratic systems, J. Diff, vol.82, pp.293-304, 1966. ,
A necessary condition for feedback stibilisation, Syst. Contr. Lett, vol.14, pp.227-232 ,
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
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
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. ,
Classification and Analysis of Two-Dimensional Real Homogeneous Quadratic Differential Equation, J . Diff. Eq, vol.32, pp.311-344, 1979. ,
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
Two examples of stabilizable second order systems, Proc. Montana Conference on Computation and Control, 1988. ,
A remark on a theorem of Andreini, Bacciotti and Stefan\. Systems ?i Control Letters, pp.363-364, 1989. ,
Asymptotic stabilization of a class of smooth two-dimensional systems, SIAM J. Control and Optimization, vol.23, pp.1321-1349, 1990. ,
Asymptotic stabilization of a generic class of three dimensional homogeneous quadratic systems, Systems I Control Letters, vol.24, pp.115-123, 1995. ,
Bounded Quadratic System in the plane, J. Difl:. Eq, vol.7, pp.251-273, 1970. ,
Partial Differential Relations, Ergebnisse Mathernatik7, F.Olge g, 1986. ,
Stabilization of planar nonlinear systems by polynomial feedback control. Systems ?i Control Letters, pp.177-185, 1994. ,
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. ,
Nilpotent and high-order approximations of vector field systems, SIAM Reaiew, vol.33, issue.2, pp.238-264, 1991. ,
Asymptotic Stabilizing Feedback Controls Journal of Differential Equati,ons,92, pp.76-89, 1991. ,
Asymptotic stabilization of planar systems, Systems ?j Control Letters, vol.17, pp.437-443 ,
Homogeneous feedback controls for homogeneous systems, Systerns ?j Control Letters, vol.24, pp.7-11, 1995. ,
Global Feedback Stabilization of Homogeneous Polynomial Systems ,
Nonlinear stabilization by adding an integrators, I{Y- BELNETII, vol.30, issue.4 5, pp.499-506, 1994. ,
Global stabilization of homogeneous polynomial systems in IR3. Nonlinear Analysis Theory, Methods ?l Apptications,lS, pp.1181-1186, 1992. ,
Nonlinear Control Systems, 1989. ,
Lagrange stability of a class of quadratic systems in lR3, Présenté à la deuxième conférence intérnationale sur les équations différentielles. Marrakech, juin 95 ,
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
Comments on the stabilization of nonlinear systems by adding an integrator, IEEE transaction on control. À paraitre ,
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. ,
Controllability and stability, Journal of Differential Equations, vol.28, issue.3, pp.381-389, 1978. ,
DOI : 10.1016/0022-0396(78)90135-3
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. ,
Homogeneous stabilizing feedback laws. Control-Theory and Adaanced Technology, pp.497-516, 1990. ,
Stabilization of nonlinear systems in the plane. Systems I Control Letters, pp.169-175, 1990. ,
Adaptive techniques for mechanical systems, Fifth Yale workshop on adaptiae systems, p.259, 1987. ,
A positive real condition for global stabilization of nonlinear systems, Systems ?i Control Letters, vol.13, pp.125-133, 1989. ,
Stability by Liapunoa's direct method with applications, 1961. ,
Quadratic differential equations and non-associative algebras, Ann. Mâth. Studi,es, pp.185-213, 1960. ,
Global stability criteria for differential systems, Oseka Math. J, vol.12, pp.305-317, 1960. ,
Contributions to Stability Theory, The Annals of Mathematics, vol.64, issue.1, pp.182-206, 1956. ,
DOI : 10.2307/1969955
Stabilisation d'une classe de systèmes affines en contrôles, Proceedings of European control conference, pp.480-484, 1991. ,
Sur Ia stabilisation globale des systèmes non linéaires par retour d'état régulier, Thèse de l'uni ,
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
Stabilization by dynamic feedback. Workshop on Nonsmooth Analysis and Differential Geometric Methods in Optimal Control. I. M. A Minneapolis, 1993. ,
Etude de quelques problemes de stabilisation. Thèse de I'uniaérsité de Paris XI, 1993. ,
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. ,
Global stabilization of partially linear composed systems, SIAM J. Control and Optirnisation, vol.23, pp.1491-1503, 1990. ,
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
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
Smooth stabilization implies coprime factorization, IEEE transaction on control N, vol.34, pp.435-443, 1989. ,
Mathematical Control Theory Deterministic finite dimentional Systems, Texts in applied Mathematics, vol.6, 1990. ,
Further comments on the stabilizability of the angular velocity of a rigid body. Systems and, Control Letters, pp.213-217, 1988. ,
Contributions to nonlinear control systems analysis by means of the direct method of Liapunov ,
The Peaking Phenomenom and the Global Stabilization of Nonlinear Systems, IEEE transaction on control,aol as, vol.36, issue.4, pp.424-440, 1991. ,