Semiconcavity results for optimal control problems admitting no singular minimizing controls

 Access to full text

 Full (PDF)

1. [1] Agrachev A., Compactness for sub-Riemannian length-minimizers and subanalyticity, Rend. Sem. Mat. Univ. Politec. Torino56 (4) (2001) 1-12. Zbl1039.53038MR1845741
2. [2] Alberti G., Ambrosio L., Cannarsa P., On the singularities of convex functions, Manuscripta Math.76 (3–4) (1992) 421-435. Zbl0784.49011MR1185029
3. [3] Bellaiche A., The tangent space in sub-Riemannian geometry, in: Sub-Riemannian Geometry, Birkhäuser, 1996, pp. 1-78. Zbl0862.53031MR1421822
4. [4] Cannarsa P., Frankowska H., Some characterizations of optimal trajectories in control theory, SIAM J. Control Optim.29 (6) (1991) 1322-1347. Zbl0744.49011MR1132185
5. [5] Cannarsa P., Pignotti C., Sinestrari C., Semiconcavity for optimal control problems with exit time, Discrete Contin. Dynam. Systems6 (4) (2000) 975-997. Zbl1009.49024MR1788264
6. [6] Cannarsa P., Sinestrari C., On a class of nonlinear time optimal control problems, Discrete Contin. Dynam. Systems1 (2) (1995) 285-300. Zbl0867.49016MR1355877
7. [7] Cannarsa P., Sinestrari C., Convexity properties of the minimum time function, Calc. Var. Partial Differential Equations3 (3) (1995) 273-298. Zbl0836.49013MR1385289
8. [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 Inc., Boston, MA, 2004. Zbl1095.49003MR2041617
9. [9] Chow C.-L., Über Systeme von linearen partiellen Differentialgleichungen ester Ordnung, Math. Ann.117 (1939) 98-105. Zbl65.0398.01MR1880JFM65.0398.01
10. [10] Clarke F.H., Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1983. Zbl0582.49001MR709590
11. [11] Clarke F., Necessary conditions in dynamic optimization, Mem. Amer. Math. Soc.173 (816) (2005). Zbl1093.49017MR2117692
12. [12] Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R., Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, vol. 178, Springer-Verlag, New York, 1998. Zbl1047.49500MR1488695
13. [13] Ge Z., Horizontal path spaces and Carnot–Carathéodory metrics, Pacific J. Math.161 (2) (1993) 255-286. Zbl0797.49033MR1242199
14. [14] I. Kupka, Géométrie sous-riemannienne, Astérisque, (241):Exp. No. 817, 5, 351–380, 1997. Séminaire Bourbaki, vol. 1995/96. Zbl0893.53013MR1472545
15. [15] Montgomery R., A Tour of Subriemannian Geometries, their Geodesics and Applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. Zbl1044.53022MR1867362
16. [16] Rashevsky P.K., About connecting two points of a completely nonholonomic space by admissible curve, Uch. Zapiski Ped. Inst. Libknechta2 (1938) 83-94. 
17. [17] Rayer C.B., The exponential map for the Lagrange problem on differentiable manifold, Philos. Trans. Roy. Soc. London Ser. A, Math. and Phys.1127 (1967) 299-344. Zbl0154.37004MR247552
18. [18] Rifford L., Existence of Lipschitz and semiconcave control-Lyapunov functions, SIAM J. Control Optim.39 (4) (2000) 1043-1064. Zbl0982.93068MR1814266
19. [19] Rifford L., Semiconcave control-Lyapunov functions and stabilizing feedbacks, SIAM J. Control Optim.41 (3) (2002) 659-681. Zbl1034.93053MR1939865
20. [20] Rifford L., The stabilization problem: AGAS and SRS feedbacks, in: Optimal Control, Stabilization, and Nonsmooth Analysis, Lectures Notes in Control and Information Sciences, vol. 301, Springer-Verlag, Heidelberg, 2004, pp. 173-184. Zbl1259.93097MR2079683
21. [21] Rifford L., A Morse–Sard theorem for the distance function on Riemannian manifolds, Manuscripta Math.113 (2004) 251-265. Zbl1051.53050MR2128549
22. [22] Rifford L., À propos des sphères sous-riemanniennes, Bull. Belg. Math. Soc. Simon Stevin13 (3) (2006) 521-526. Zbl1135.53021MR2307687
23. [23] L. Rifford, E. Trélat, On the stabilization problem for nonholonomic distributions, J. Eur. Math. Soc., in press. Zbl1194.49070
24. [24] Trélat E., Some properties of the value function and its level sets for affine control systems with quadratic cost, J. Dynamical Control Systems6 (4) (2000) 511-541. Zbl0964.49021MR1778212
25. [25] Weinstein A., Fat bundles and symplectic manifolds, Adv. in Math.37 (1980) 239-250. Zbl0449.53035MR591728