# American Institute of Mathematical Sciences

September  2013, 3(3): 303-322. doi: 10.3934/mcrf.2013.3.303

## Phantom tracking method, homogeneity and rapid stabilization

 1 Institut universitaire de France and Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France

Received  January 2013 Revised  April 2013 Published  September 2013

In this paper we explain on various examples the phantom tracking'' method, a method which can be used to stabilize nonlinear control systems modeled by ordinary differential equations or partial differential equations. We show how it can handle global controllability, homogeneity issues or fast stabilization.
Citation: Jean-Michel Coron. Phantom tracking method, homogeneity and rapid stabilization. Mathematical Control and Related Fields, 2013, 3 (3) : 303-322. doi: 10.3934/mcrf.2013.3.303
##### References:
 [1] Karine Beauchard, Jean Michel Coron, Mazyar Mirrahimi and Pierre Rouchon, Implicit lyapunov control of finite dimensional schrödinger equations, Systems Control Lett., 56 (2007), 388-395. doi: 10.1016/j.sysconle.2006.10.024. [2] Sanjay P. Bhat and Dennis S. Bernstein, Geometric homogeneity with applications to finite-time stability, Math. Control Signals Systems, 17 (2005), 101-127. doi: 10.1007/s00498-005-0151-x. [3] Bernard Bonnard, Contrôlabilité de systemes mécaniques sur les groupes de lie, (French) [Controllability of mechanical systems on Lie groups] SIAM J. Control Optim., 22 (1984), 711-722. doi: 10.1137/0322045. [4] Bernard Bonnard, Ludovic Faubourg and Emmanuel Trélat, "Mécanique CÉleste Et Contrôle Des VÉhicules Spatiaux," (French) [Celestial mechanics and the control of space vehicles] Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006. [5] Christopher I. Byrnes and Alberto Isidori, New results and examples in nonlinear feedback stabilization, Systems Control Lett., 12 (1989), 437-442. doi: 10.1016/0167-6911(89)90080-7. [6] Jean-Michel Coron, On the controllability of $2$-D incompressible perfect fluids, J. Math. Pures Appl., 75 (1996), 155-188. [7] Jean-Michel Coron, On the null asymptotic stabilization of the two-dimensional incompressible euler equations in a simply connected domain, SIAM J. Control Optim., 37 (1999), 1874-1896. doi: 10.1137/S036301299834140X. [8] Jean-Michel Coron, Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations, A tribute to J. L. Lions. ESAIM Control Optim. Calc. Var., 8 (2002), 513-554. doi: 10.1051/cocv:2002050. [9] Jean-Michel Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007. [10] Jean-Michel Coron, On the controllability of nonlinear partial differential equations, In "Proceedings of the International Congress of Mathematicians. Volume I," pages 238-264, Hindustan Book Agency, New Delhi, 2010. [11] Wijesuriya P. Dayawansa, Clyde F. Martin and Sandra L. Samelson Samelson, Asymptotic stabilization of a generic class of three-dimensional homogeneous quadratic systems, Systems Control Lett., 24 (1995), 115-123. doi: 10.1016/0167-6911(94)00040-3. [12] Ludovic Faubourg and Jean-Baptiste Pomet, Control lyapunov functions for homogeneous "Jurdjevic-Quinn'' systems, ESAIM Control Optim. Calc. Var., 5 (2000), 293-311(electronic). doi: 10.1051/cocv:2000112. [13] Olivier Glass, Exact boundary controllability of 3-D euler equation, ESAIM Control Optim. Calc. Var., 5 (2000), 1-44 (electronic). doi: 10.1051/cocv:2000100. [14] Olivier Glass, Asymptotic stabilizability by stationary feedback of the two-dimensional euler equation: The multiconnected case, SIAM J. Control Optim., 44 (2005), 1105-1147 (electronic). doi: 10.1137/S0363012903431153. [15] Andreea Grigoriu, Implicit lyapunov control for schrödinger equations with dipole and polarizability term, Proc. of the 50th IEEE Conf. on Decision and Control, USA, December 2011. [16] Henry Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, In "Differential Equations" (Colorado Springs, CO, 1989), Lecture Notes in Pure and Appl. Math., pages 249-260. Dekker, New York, 1991. [17] Henry Hermes, Smooth homogeneous asymptotically stabilizing feedback controls, ESAIM Control Optim. Calc. Var., 2 (1997), 13-32 (electronic). doi: 10.1051/cocv:1997101. [18] Hamadi Jerbi, On the stabilizability of homogeneous systems of odd degree, ESAIM Control Optim. Calc. Var., 9 (2003), 343-352 (electronic). doi: 10.1051/cocv:2003016. [19] Hassan K. Khalil, "Nonlinear Systems," Prentice Hall, New Jersey, 2002. doi: 10.1007/s11071-008-9349-z. [20] Daniel E. Koditschek, Adaptive techniques for mechanical systems, In "Proc. 5th. Yale University Conference, New Haven" (1987), pages 259-265. Yale University, New Haven, 1987. [21] Pascal Morin, Jean-Baptiste Pomet and Claude Samson, Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of lie brackets in closed loop, SIAM J. Control Optim., 38 (1999), 22-49 (electronic). doi: 10.1137/S0363012997315427. [22] Pascal Morin and Claude Samson, Time-varying exponential stabilization of a rigid spacecraft with two control torques, IEEE Trans. Automat. Control, 42 (1997), 528-534. doi: 10.1109/9.566663. [23] Laurent Praly, Brigitte d'Andréa-Novel and Jean-Michel Coron, Lyapunov design of stabilizing controllers for cascaded systems, IEEE Trans. Automat. Control, 36 (1991), 1177-1181. doi: 10.1109/9.90230. [24] Lionel Rosier, Homogeneous lyapunov function for homogeneous continuous vector fields, Systems Control Lett., 19 (1992), 467-473. doi: 10.1016/0167-6911(92)90078-7. [25] Eduardo D. Sontag, "Mathematical Control Theory," Deterministic finite-dimensional systems, Second edition, Texts in Applied Mathematics, Springer-Verlag, New York, 1998. [26] John Tsinias, Sufficient Lyapunov-like conditions for stabilization, Math. Control Signals Systems, 2 (1989), 343-357. doi: 10.1007/BF02551276.

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##### References:
 [1] Karine Beauchard, Jean Michel Coron, Mazyar Mirrahimi and Pierre Rouchon, Implicit lyapunov control of finite dimensional schrödinger equations, Systems Control Lett., 56 (2007), 388-395. doi: 10.1016/j.sysconle.2006.10.024. [2] Sanjay P. Bhat and Dennis S. Bernstein, Geometric homogeneity with applications to finite-time stability, Math. Control Signals Systems, 17 (2005), 101-127. doi: 10.1007/s00498-005-0151-x. [3] Bernard Bonnard, Contrôlabilité de systemes mécaniques sur les groupes de lie, (French) [Controllability of mechanical systems on Lie groups] SIAM J. Control Optim., 22 (1984), 711-722. doi: 10.1137/0322045. [4] Bernard Bonnard, Ludovic Faubourg and Emmanuel Trélat, "Mécanique CÉleste Et Contrôle Des VÉhicules Spatiaux," (French) [Celestial mechanics and the control of space vehicles] Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006. [5] Christopher I. Byrnes and Alberto Isidori, New results and examples in nonlinear feedback stabilization, Systems Control Lett., 12 (1989), 437-442. doi: 10.1016/0167-6911(89)90080-7. [6] Jean-Michel Coron, On the controllability of $2$-D incompressible perfect fluids, J. Math. Pures Appl., 75 (1996), 155-188. [7] Jean-Michel Coron, On the null asymptotic stabilization of the two-dimensional incompressible euler equations in a simply connected domain, SIAM J. Control Optim., 37 (1999), 1874-1896. doi: 10.1137/S036301299834140X. [8] Jean-Michel Coron, Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations, A tribute to J. L. Lions. ESAIM Control Optim. Calc. Var., 8 (2002), 513-554. doi: 10.1051/cocv:2002050. [9] Jean-Michel Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007. [10] Jean-Michel Coron, On the controllability of nonlinear partial differential equations, In "Proceedings of the International Congress of Mathematicians. Volume I," pages 238-264, Hindustan Book Agency, New Delhi, 2010. [11] Wijesuriya P. Dayawansa, Clyde F. Martin and Sandra L. Samelson Samelson, Asymptotic stabilization of a generic class of three-dimensional homogeneous quadratic systems, Systems Control Lett., 24 (1995), 115-123. doi: 10.1016/0167-6911(94)00040-3. [12] Ludovic Faubourg and Jean-Baptiste Pomet, Control lyapunov functions for homogeneous "Jurdjevic-Quinn'' systems, ESAIM Control Optim. Calc. Var., 5 (2000), 293-311(electronic). doi: 10.1051/cocv:2000112. [13] Olivier Glass, Exact boundary controllability of 3-D euler equation, ESAIM Control Optim. Calc. Var., 5 (2000), 1-44 (electronic). doi: 10.1051/cocv:2000100. [14] Olivier Glass, Asymptotic stabilizability by stationary feedback of the two-dimensional euler equation: The multiconnected case, SIAM J. Control Optim., 44 (2005), 1105-1147 (electronic). doi: 10.1137/S0363012903431153. [15] Andreea Grigoriu, Implicit lyapunov control for schrödinger equations with dipole and polarizability term, Proc. of the 50th IEEE Conf. on Decision and Control, USA, December 2011. [16] Henry Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, In "Differential Equations" (Colorado Springs, CO, 1989), Lecture Notes in Pure and Appl. Math., pages 249-260. Dekker, New York, 1991. [17] Henry Hermes, Smooth homogeneous asymptotically stabilizing feedback controls, ESAIM Control Optim. Calc. Var., 2 (1997), 13-32 (electronic). doi: 10.1051/cocv:1997101. [18] Hamadi Jerbi, On the stabilizability of homogeneous systems of odd degree, ESAIM Control Optim. Calc. Var., 9 (2003), 343-352 (electronic). doi: 10.1051/cocv:2003016. [19] Hassan K. Khalil, "Nonlinear Systems," Prentice Hall, New Jersey, 2002. doi: 10.1007/s11071-008-9349-z. [20] Daniel E. Koditschek, Adaptive techniques for mechanical systems, In "Proc. 5th. Yale University Conference, New Haven" (1987), pages 259-265. Yale University, New Haven, 1987. [21] Pascal Morin, Jean-Baptiste Pomet and Claude Samson, Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of lie brackets in closed loop, SIAM J. Control Optim., 38 (1999), 22-49 (electronic). doi: 10.1137/S0363012997315427. [22] Pascal Morin and Claude Samson, Time-varying exponential stabilization of a rigid spacecraft with two control torques, IEEE Trans. Automat. Control, 42 (1997), 528-534. doi: 10.1109/9.566663. [23] Laurent Praly, Brigitte d'Andréa-Novel and Jean-Michel Coron, Lyapunov design of stabilizing controllers for cascaded systems, IEEE Trans. Automat. Control, 36 (1991), 1177-1181. doi: 10.1109/9.90230. [24] Lionel Rosier, Homogeneous lyapunov function for homogeneous continuous vector fields, Systems Control Lett., 19 (1992), 467-473. doi: 10.1016/0167-6911(92)90078-7. [25] Eduardo D. Sontag, "Mathematical Control Theory," Deterministic finite-dimensional systems, Second edition, Texts in Applied Mathematics, Springer-Verlag, New York, 1998. [26] John Tsinias, Sufficient Lyapunov-like conditions for stabilization, Math. Control Signals Systems, 2 (1989), 343-357. doi: 10.1007/BF02551276.
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