On the existence of local smooth repulsive stabilizing feedbacks in dimension three

Journal of Differential Equations

Volumn 226, Issue 2, 2006, Pages 429-500

  Rifford, Ludovic ^a  

^a Université Paris Sud 11   (France)

Author keywords

Control theory; Control Lyapunov functions; Homogeneous systems; Stabilization problem

Indexed keywords

EID: 33646501704     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2005.10.017     Document Type: Article

References (47)

1. Ancona F., and Bressan A. Patchy vector fields and asymptotic stabilization. ESAIM Control Optim. Calc. Var. 4 (1999) 445-471
2. Ancona F., and Bressan A. Flow stability of patchy vector fields and robust feedback stabilization. SIAM J. Control Optim. 41 5 (2002) 1455-1476
3. Aubin J.-P., and Cellina A. Differential Inclusions. Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences) vol. 264 (1984), Springer, Berlin
4. Artstein Z. Stabilization with relaxed controls. Nonlinear Anal. 7 (1983) 1163-1173
5. Bardi M., and Capuzzo-Dolcetta I. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems Control Found. Appl. (1997), Birkhäuser Boston, Boston, MA
6. Bellaiche A. The tangent space in sub-Riemannian geometry. Sub-Riemannian Geometry (1996), Birkhäuser, Basel 1-78
7. Bredon G.E. Topology and Geometry. Grad. Texts in Math. vol. 139 (1993), Springer, New York
8. Bressan A. Local asymptotic approximation of nonlinear control systems. Internat. J. Control 41 5 (1985) 1331-1336
9. Brockett R.W. Asymptotic stability and feedback stabilization. In: Brockett R.W., Millman R.S., and Sussmann H.J. (Eds). Differential Geometric Control Theory (1983), Birkhäuser Boston, Boston 181-191
10. Cannarsa P., and Sinestrari C. Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations and Their Applications vol. 58 (2004), Birkhäuser Boston, Boston, MA
11. Chow C.-L. Über Systeme von linearen partiellen Differentialgleichungen ester Ordnung. Math. Ann. 117 (1939) 98-105
12. Clarke F.H. Optimization and Nonsmooth Analysis (1983), Wiley-Interscience, New York republished as
13. Clarke F.H. Optimization and Nonsmooth Analysis. Classics in Appl. Math. vol. 5 (1990), SIAM
14. Clarke F.H., Ledyaev Yu.S., Rifford L., and Stern R.J. Feedback stabilization and Lyapunov functions. SIAM J. Control Optim. 39 1 (2000) 25-48
15. Clarke F.H., Ledyaev Yu.S., Sontag E.D., and Subbotin A.I. Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control 42 (1997) 1394-1407
16. Clarke F.H., Ledyaev Yu.S., Stern R.J., and Wolenski P.R. Nonsmooth Analysis and Control Theory. Grad. Texts in Math. vol. 178 (1998), Springer, New York
17. Coron J.-M. A necessary condition for feedback stabilization. Systems Control Lett. 14 (1990) 227-232
18. Coron J.-M. Global asymptotic stabilization of controllable systems without drift. Math. Control Signals Systems 5 (1992) 295-312
19. Coron J.-M. Linearized control systems and application to smooth stabilization. SIAM J. Control Optim. 32 2 (1994) 358-386
20. Coron J.-M. On the stabilization of some nonlinear control systems: Results, tools, and applications. Nonlinear Analysis, Differential Equations and Control. Montreal, QC, 1998 (1999), Kluwer Academic, Dordrecht 307-367
21. Deimling K. Multivalued Differential Equations. de Gruyter Ser. Nonlinear Anal. Appl. vol. 1 (1992), de Gruyter, Berlin
22. Grüne L. Homogeneous state feedback stabilization of homogeneous systems. SIAM J. Control Optim. 38 4 (2000) 1288-1308
23. Hermes H. Nilpotent and high-order approximations of vector field systems. SIAM Rev. 33 2 (1991) 238-264
24. Hermes H. Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. In: Elaydi S. (Ed). Differential Equations, Stability and Control. Lecture Notes in Pure and Appl. Math. vol. 127 (1991), Dekker, New York 249-260
25. Hironaka H. Introduction aux ensembles sous-analytiques. Astérisque 7-8 (1973) 13-20
26. Hironaka H. Subanalytic sets. Number Theory, Algebraic Geometry and Commutative Algebra, in Honor of Y. Akizuki (1973), Kinokuniya, Tokyo
27. Hirsch M., and Smale S. Differential Equations, Dynamical Systems, and Linear Algebra. Pure Appl. Math. vol. 60 (1974), Academic Press, New York/London
28. S. Lojasiewicz, Ensembles semi-analytiques, Lecture Note at IHES, Bures-sur-Yvette; Reproduit NA66.765, École polytechnique, Paris, 1965
29. Michael E. Continuous selections. I. Ann. of Math. (2) 63 (1956) 361-382
30. Morin P., Pomet J.-B., and Samson C. 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 1 (1999) 22-49
31. Prieur C. Asymptotic controllability and robust asymptotic stabilizability. SIAM J. Control Optim. 43 5 (2005) 1888-1912
32. Rashevsky P.K. About connecting two points of a completely nonholonomic space by admissible curve. Uch. Zapiski Ped. Inst. Libknechta 2 (1938) 83-94
33. L. Rifford, Problèmes de stabilisation en théorie du controle, PhD thesis, Université Claude Bernard Lyon I, 2000
34. Rifford L. Stabilisation des systèmes globalement asymptotiquement commandables. C. R. Acad. Sci. Paris Sér. I Math. 330 3 (2000) 211-216
35. Rifford L. Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J. Control Optim. 39 4 (2000) 1043-1064
36. Rifford L. On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients. ESAIM Control Optim. Calc. Var. 6 (2001) 593-611
37. Rifford L. Semiconcave control-Lyapunov functions and stabilizing feedbacks. SIAM J. Control Optim. 41 3 (2002) 659-681
38. Rifford L. Singularities of viscosity solutions and the stabilization problem in the plane. Indiana Univ. Math. J. 52 5 (2003) 1373-1396
39. Rifford L. The stabilization problem: AGAS and SRS feedbacks. Optimal Control, Stabilization, and Nonsmooth Analysis. Lectures Notes in Control and Inform. Sci. vol. 301 (2004), Springer, Heidelberg 173-184
40. L. Rifford, On the stabilization problem on surfaces, Rend. Semin. Mat. Torino, in press
41. Rifford L. Stratified semiconcave control-Lyapunov functions and the stabilization problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 3 (2005) 343-384
42. Rosier L. Homogeneous Lyapunov function for homogeneous continuous vector field. Systems Control Lett. 19 6 (1992) 467-473
43. Rothschild L.P., and Stein E.M. Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976) 247-320
44. Shiota M. Geometry of Subanalytic and Semialgebraic Sets. Progr. Math. vol. 150 (1997), Birkhaüser Boston, Boston
45. Sontag E.D. A Lyapunov-like characterization of asymptotic controllability. SIAM J. Control Optim. 21 (1983) 462-471
46. Sontag E.D. Stability and stabilization: discontinuities and the effect of disturbances. Nonlinear Analysis, Differential Equations and Control. Montreal, QC, 1998 (1999), Kluwer Academic, Dordrecht 307-367
47. Sussmann H.J. Subanalytic sets and feedback control. J. Differential Equations 31 1 (1979) 31-52