Stratified semiconcave control-Lyapunov functions and the stabilization problem

Rifford, Ludovic

Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 3, pp. 343-384.

[1] Albano P., Cannarsa P., Structural properties of singularities of semiconcave functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (4) (1999) 719-740. | Numdam | MR | Zbl

[2] Alberti G., Ambrosio L., Cannarsa P., On the singularities of convex functions, Manuscripta Math. 76 (3-4) (1992) 421-435. | MR | Zbl

[3] Ancona F., Bressan A., Patchy vector fields and asymptotic stabilization, ESAIM Control Optim. Calc. Var. 4 (1999) 445-471. | Numdam | MR | Zbl

[4] Astolfi A., Discontinuous control of nonholonomic systems, Systems Control Lett. 27 (1) (1996) 37-45. | MR | Zbl

[5] Aubin J.-P., Viability Theory, Birkhäuser Boston, Boston MA, 1991. | MR | Zbl

[6] Bardi M., Capuzzo-Dolcetta I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Boston, MA, 1997. | MR | Zbl

[7] Brockett R.W., Asymptotic stability and feedback stabilization, in: Brockett R.W., Millman R.S., Sussmann H.J. (Eds.), Differential Geometric Control Theory, Birkhäuser, Boston, 1983, pp. 181-191. | MR | Zbl

[8] Cannarsa P., Sinestrari C., Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and Their Applications, vol. 58, Birkhäuser Boston, Boston, MA, 2004. | MR | Zbl

[9] Clarke F.H., Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983. | MR | Zbl

[10] Clarke F.H., Ledyaev Yu.S., Sontag E.D., Subbotin A.I., Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control 42 (1997) 1394-1407. | MR | Zbl

[11] Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R., Nonsmooth Analysis and Control Theory, Graduate Texts in Math., vol. 178, Springer-Verlag, New York, 1998. | MR | Zbl

[12] Clarke F.H., Stern R.J., Wolenski P.R., Proximal smoothness and the lower-$C^2$ property, J. Convex Anal. 2 (1995) 117-145. | MR | Zbl

[13] Coron J.-M., On the stabilization of some nonlinear control systems: results, tools, and applications, in: Nonlinear Analysis, Differential Equations and Control (Montreal, QC, 1998), Kluwer Academic, Dordrecht, 1999, pp. 307-367. | MR | Zbl

[14] Crandall M.G., Lions P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1) (1983) 1-42. | MR | Zbl

[15] Gantmacher F.R., The Theory of Matrices. Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998. | MR | Zbl

[16] Goresky M., Macpherson R., Stratified Morse Theory, Springer-Verlag, Berlin, 1988. | MR | Zbl

[17] Lions P.-L., Generalized Solutions of Hamilton-Jacobi Equations, Pitman, Boston, MA, 1982, (Advanced Publishing Program). | MR | Zbl

[18] Michael E., Continuous selections. I, Ann. of Math. (2) 63 (1956) 361-382. | MR | Zbl

[19] Rantzer A., A dual to Lyapunov stability theorem, Systems Control Lett. 42 (3) (2000) 161-168. | MR | Zbl

[20] A. Rantzer, A converse Lyapunov stability theorem, Personnal communication.

[21] L. Rifford, Problèmes de stabilisation en théorie du controle, PhD thesis, Université Claude Bernard Lyon I, 2000.

[22] Rifford L., Stabilisation des systèmes globalement asymptotiquement commandables, C. R. Acad. Sci. Paris Sér. I Math. 330 (3) (2000) 211-216. | MR | Zbl

[23] Rifford L., Existence of Lipschitz and semiconcave control-Lyapunov functions, SIAM J. Control Optim. 39 (4) (2000) 1043-1064. | MR | Zbl

[24] Rifford L., Semiconcave control-Lyapunov functions and stabilizing feedbacks, SIAM J. Control Optim. 41 (3) (2002) 659-681. | MR | Zbl

[25] Rifford L., Singularities of viscosity solutions and the stabilization problem in the plane, Indiana Univ. Math. J. 52 (5) (2003) 1373-1396. | MR | Zbl

[26] Rifford L., A Morse-Sard theorem for the distance function on Riemannian manifolds, Manuscripta Math. 113 (2004) 251-265. | MR | Zbl

[27] L. Rifford, On the existence of local smooth repulsive stabilizing feedbacks in dimension three, in preparation. | Zbl

[28] L. Rifford, The stabilization problem on surfaces, Rend. Semin. Mat. Torino, submitted for publication.

[29] Rockafellar R.T., Convex Analysis, Princeton University Press, Princeton, NJ, 1997, Reprint of the 1970 original, Princeton Paperbacks. | MR | Zbl

[30] Sontag E.D., A Lyapunov-like characterization of asymptotic controllability, SIAM J. Control Optim. 21 (1983) 462-471. | MR | Zbl

[31] Sontag E.D., Stability and stabilization: discontinuities and the effect of disturbances, in: Nonlinear Analysis, Differential Equations and Control (Montreal, QC, 1998), Kluwer Academic, Dordrecht, 1999, pp. 307-367. | MR | Zbl

[32] Sontag E.D., Clocks and insensitivity to small measurement errors, ESAIM Control Optim. Calc. Var. 4 (1999) 537-557. | Numdam | MR | Zbl

[33] Sussmann H.J., Subanalytic sets and feedback control, J. Differential Equations 31 (1) (1979) 31-52. | MR | Zbl