Regular syntheses and solutions to discontinuous ODEs
ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002) , pp. 291-307.

In this paper we analyze several concepts of solution to discontinuous ODEs in relation to feedbacks generated by optimal syntheses. Optimal trajectories are called Stratified Solutions in case of regular synthesis in the sense of Boltyanskii-Brunovsky. We introduce a concept of solution called Krasowskii Cone Robust that characterizes optimal trajectories for minimum time on the plane and for general problems under suitable assumptions.

DOI : https://doi.org/10.1051/cocv:2002013
Classification : 49K15,  34A36,  93C15
Mots clés : optimal control, regular synthesis, discontinuous ODEs
@article{COCV_2002__7__291_0,
     author = {Marigo, Alessia and Piccoli, Benedetto},
     title = {Regular syntheses and solutions to discontinuous ODEs},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {291--307},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2002},
     doi = {10.1051/cocv:2002013},
     zbl = {1053.49027},
     mrnumber = {1925031},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2002013/}
}
Marigo, Alessia; Piccoli, Benedetto. Regular syntheses and solutions to discontinuous ODEs. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002) , pp. 291-307. doi : 10.1051/cocv:2002013. http://www.numdam.org/articles/10.1051/cocv:2002013/

[1] J.P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag (1984). | MR 755330 | Zbl 0538.34007

[2] A. Bicchi, A. Ballucchi, B. Piccoli and P. Soueres, Stability and robustness of optimal synthesis for route tracking by dubins' vehicles, in Proc. IEEE Int. Conf. on Decision and Control (2000).

[3] V.G. Boltyanskii, Sufficient conditions for optimality and the justification of the dynamic programming principle. SIAM J. Control Optim. 4 (1966) 326-361. | MR 197205 | Zbl 0143.32004

[4] U. Boscain and B. Piccoli, Extremal synthesis for generic planar systems. J. Dynam. Control Systems 7 (2001) 209-258. | MR 1830492 | Zbl 1013.49025

[5] A. Bressan and B. Piccoli, A generic classification of time-optimal planar stabilizing feedbacks. SIAM J. Control Optim. 36 (1998) 12-32. | MR 1616525 | Zbl 0910.93044

[6] P. Brunovsky, Existence of regular syntheses for general problems. J. Differential Equations 38 (1980) 317-343. | MR 605053 | Zbl 0417.49030

[7] F.H. Clarke, Y.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control 42 (1997) 1394-1407. | MR 1472857 | Zbl 0892.93053

[8] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and Its Applications (Soviet Series). Kluwer Academic Publishers Group (1988). Translated from the Russian. | MR 1028776 | Zbl 0664.34001

[9] C.G. Gibson, Construction of canonical stratifications, in Topological Stability of Smooth Mappings. Springer Verlag, Berlin, Lecture Notes in Math. 552 (1976) 9-34. | MR 436203

[10] M. Goresky and R. Macpherson, Stratified Morse Theory. Springer Verlag, Berlin (1988). | MR 932724 | Zbl 0639.14012

[11] A.J. Krener and H. Schättler, The structure of small time reachable sets in low dimensions. SIAM J. Control Optim. 27 (1989) 120-147. | MR 980227 | Zbl 0669.49020

[12] J.P. Laumond and P. Souères, Shortest paths synthesis for a car-like robot. IEEE Trans. Automat. Control 41 (1996) 672-688. | MR 1386992 | Zbl 0864.93076

[13] A. Marigo and B. Piccoli, Safety controls and applications to the dubins' car. Nonlinear Differential Equations and Applications (in print). | Zbl 1051.49021

[14] A. Marigo and B. Piccoli, Safety driving for the dubins' car, in XV World Congress on Automatic Control b'02 (in print).

[15] R. Murray, Nonlinear control of mechanical systems: A Lagrangian perspective, in IFAC Symposium on Nonlinear Control Systems Design (NOLCOS) (1995) 378-389.

[16] B. Piccoli, Classification of generic singularities for the planar time optimal syntheses. SIAM J. Control Optim. 34 (1996) 914-1946. | MR 1416494 | Zbl 0865.49022

[17] B. Piccoli and H.J. Sussmann, Regular synthesis and sufficiency conditions for optimality. SIAM J. Control Optim. 39 (2000) 359-410. | MR 1788064 | Zbl 0961.93014

[18] H.J. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane. SIAM J. Control Optim. 25 (1987) 1145-1162. | MR 905037 | Zbl 0696.93026