# 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 & 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. doi: 10.1016/j.sysconle.2006.10.024. Google Scholar [2] Sanjay P. Bhat and Dennis S. Bernstein, Geometric homogeneity with applications to finite-time stability,, Math. Control Signals Systems, 17 (2005), 101. doi: 10.1007/s00498-005-0151-x. Google Scholar [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. doi: 10.1137/0322045. Google Scholar [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, (2006). Google Scholar [5] Christopher I. Byrnes and Alberto Isidori, New results and examples in nonlinear feedback stabilization,, Systems Control Lett., 12 (1989), 437. doi: 10.1016/0167-6911(89)90080-7. Google Scholar [6] Jean-Michel Coron, On the controllability of $2$-D incompressible perfect fluids,, J. Math. Pures Appl., 75 (1996), 155. Google Scholar [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. doi: 10.1137/S036301299834140X. Google Scholar [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. doi: 10.1051/cocv:2002050. Google Scholar [9] Jean-Michel Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs,, American Mathematical Society, (2007). Google Scholar [10] Jean-Michel Coron, On the controllability of nonlinear partial differential equations,, In, (2010), 238. Google Scholar [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. doi: 10.1016/0167-6911(94)00040-3. Google Scholar [12] Ludovic Faubourg and Jean-Baptiste Pomet, Control lyapunov functions for homogeneous "Jurdjevic-Quinn'' systems,, ESAIM Control Optim. Calc. Var., 5 (2000), 293. doi: 10.1051/cocv:2000112. Google Scholar [13] Olivier Glass, Exact boundary controllability of 3-D euler equation,, ESAIM Control Optim. Calc. Var., 5 (2000), 1. doi: 10.1051/cocv:2000100. Google Scholar [14] Olivier Glass, Asymptotic stabilizability by stationary feedback of the two-dimensional euler equation: The multiconnected case,, SIAM J. Control Optim., 44 (2005), 1105. doi: 10.1137/S0363012903431153. Google Scholar [15] Andreea Grigoriu, Implicit lyapunov control for schrödinger equations with dipole and polarizability term,, Proc. of the 50th IEEE Conf. on Decision and Control, (2011). Google Scholar [16] Henry Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls,, In, (1989), 249. Google Scholar [17] Henry Hermes, Smooth homogeneous asymptotically stabilizing feedback controls,, ESAIM Control Optim. Calc. Var., 2 (1997), 13. doi: 10.1051/cocv:1997101. Google Scholar [18] Hamadi Jerbi, On the stabilizability of homogeneous systems of odd degree,, ESAIM Control Optim. Calc. Var., 9 (2003), 343. doi: 10.1051/cocv:2003016. Google Scholar [19] Hassan K. Khalil, "Nonlinear Systems,", Prentice Hall, (2002). doi: 10.1007/s11071-008-9349-z. Google Scholar [20] Daniel E. Koditschek, Adaptive techniques for mechanical systems,, In, (1987), 259. Google Scholar [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. doi: 10.1137/S0363012997315427. Google Scholar [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. doi: 10.1109/9.566663. Google Scholar [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. doi: 10.1109/9.90230. Google Scholar [24] Lionel Rosier, Homogeneous lyapunov function for homogeneous continuous vector fields,, Systems Control Lett., 19 (1992), 467. doi: 10.1016/0167-6911(92)90078-7. Google Scholar [25] Eduardo D. Sontag, "Mathematical Control Theory," Deterministic finite-dimensional systems, Second edition, Texts in Applied Mathematics,, Springer-Verlag, (1998). Google Scholar [26] John Tsinias, Sufficient Lyapunov-like conditions for stabilization,, Math. Control Signals Systems, 2 (1989), 343. doi: 10.1007/BF02551276. Google Scholar

show all references

##### 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. doi: 10.1016/j.sysconle.2006.10.024. Google Scholar [2] Sanjay P. Bhat and Dennis S. Bernstein, Geometric homogeneity with applications to finite-time stability,, Math. Control Signals Systems, 17 (2005), 101. doi: 10.1007/s00498-005-0151-x. Google Scholar [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. doi: 10.1137/0322045. Google Scholar [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, (2006). Google Scholar [5] Christopher I. Byrnes and Alberto Isidori, New results and examples in nonlinear feedback stabilization,, Systems Control Lett., 12 (1989), 437. doi: 10.1016/0167-6911(89)90080-7. Google Scholar [6] Jean-Michel Coron, On the controllability of $2$-D incompressible perfect fluids,, J. Math. Pures Appl., 75 (1996), 155. Google Scholar [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. doi: 10.1137/S036301299834140X. Google Scholar [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. doi: 10.1051/cocv:2002050. Google Scholar [9] Jean-Michel Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs,, American Mathematical Society, (2007). Google Scholar [10] Jean-Michel Coron, On the controllability of nonlinear partial differential equations,, In, (2010), 238. Google Scholar [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. doi: 10.1016/0167-6911(94)00040-3. Google Scholar [12] Ludovic Faubourg and Jean-Baptiste Pomet, Control lyapunov functions for homogeneous "Jurdjevic-Quinn'' systems,, ESAIM Control Optim. Calc. Var., 5 (2000), 293. doi: 10.1051/cocv:2000112. Google Scholar [13] Olivier Glass, Exact boundary controllability of 3-D euler equation,, ESAIM Control Optim. Calc. Var., 5 (2000), 1. doi: 10.1051/cocv:2000100. Google Scholar [14] Olivier Glass, Asymptotic stabilizability by stationary feedback of the two-dimensional euler equation: The multiconnected case,, SIAM J. Control Optim., 44 (2005), 1105. doi: 10.1137/S0363012903431153. Google Scholar [15] Andreea Grigoriu, Implicit lyapunov control for schrödinger equations with dipole and polarizability term,, Proc. of the 50th IEEE Conf. on Decision and Control, (2011). Google Scholar [16] Henry Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls,, In, (1989), 249. Google Scholar [17] Henry Hermes, Smooth homogeneous asymptotically stabilizing feedback controls,, ESAIM Control Optim. Calc. Var., 2 (1997), 13. doi: 10.1051/cocv:1997101. Google Scholar [18] Hamadi Jerbi, On the stabilizability of homogeneous systems of odd degree,, ESAIM Control Optim. Calc. Var., 9 (2003), 343. doi: 10.1051/cocv:2003016. Google Scholar [19] Hassan K. Khalil, "Nonlinear Systems,", Prentice Hall, (2002). doi: 10.1007/s11071-008-9349-z. Google Scholar [20] Daniel E. Koditschek, Adaptive techniques for mechanical systems,, In, (1987), 259. Google Scholar [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. doi: 10.1137/S0363012997315427. Google Scholar [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. doi: 10.1109/9.566663. Google Scholar [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. doi: 10.1109/9.90230. Google Scholar [24] Lionel Rosier, Homogeneous lyapunov function for homogeneous continuous vector fields,, Systems Control Lett., 19 (1992), 467. doi: 10.1016/0167-6911(92)90078-7. Google Scholar [25] Eduardo D. Sontag, "Mathematical Control Theory," Deterministic finite-dimensional systems, Second edition, Texts in Applied Mathematics,, Springer-Verlag, (1998). Google Scholar [26] John Tsinias, Sufficient Lyapunov-like conditions for stabilization,, Math. Control Signals Systems, 2 (1989), 343. doi: 10.1007/BF02551276. Google Scholar
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