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Regular syntheses and solutions to discontinuous ODEs

Published online by Cambridge University Press:  15 September 2002

Alessia Marigo  and
Benedetto Piccoli
Alessia Marigo
Affiliation:
IAC-CNR, Viale Policlinico 137, 00161 Roma, Italy; marigo@iac.rm.cnr.it..; piccoli@iac.rm.cnr.it..
Benedetto Piccoli
Affiliation:
IAC-CNR, Viale Policlinico 137, 00161 Roma, Italy; marigo@iac.rm.cnr.it..; piccoli@iac.rm.cnr.it..
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Abstract

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.

Keywords

Optimal controlregular synthesisdiscontinuous ODEs.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 291 - 307
Copyright
© EDP Sciences, SMAI, 2002

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References

J.P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag (1984).
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).
Boltyanskii, V.G., Sufficient conditions for optimality and the justification of the dynamic programming principle. SIAM J. Control Optim. 4 (1966) 326-361. CrossRef
Boscain, U. and Piccoli, B., Extremal synthesis for generic planar systems. J. Dynam. Control Systems 7 (2001) 209-258. CrossRef
Bressan, A. and Piccoli, B., A generic classification of time-optimal planar stabilizing feedbacks. SIAM J. Control Optim. 36 (1998) 12-32. CrossRef
Brunovsky, P., Existence of regular syntheses for general problems. J. Differential Equations 38 (1980) 317-343. CrossRef
Clarke, F.H., Ledyaev, Y.S., Sontag, E.D. and Subbotin, A.I., Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control 42 (1997) 1394-1407. CrossRef
A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and Its Applications (Soviet Series). Kluwer Academic Publishers Group (1988). Translated from the Russian.
Gibson, C.G., Construction of canonical stratifications, in Topological Stability of Smooth Mappings. Springer Verlag, Berlin, Lecture Notes in Math. 552 (1976) 9-34.
M. Goresky and R. MacPherson, Stratified Morse Theory. Springer Verlag, Berlin (1988).
Krener, A.J. and Schättler, H., The structure of small time reachable sets in low dimensions. SIAM J. Control Optim. 27 (1989) 120-147. CrossRef
Laumond, J.P. and Souères, P., Shortest paths synthesis for a car-like robot. IEEE Trans. Automat. Control 41 (1996) 672-688.
A. Marigo and B. Piccoli, Safety controls and applications to the dubins' car. Nonlinear Differential Equations and Applications (in print).
A. Marigo and B. Piccoli, Safety driving for the dubins' car, in XV World Congress on Automatic Control b'02 (in print).
R. Murray, Nonlinear control of mechanical systems: A Lagrangian perspective, in IFAC Symposium on Nonlinear Control Systems Design (NOLCOS) (1995) 378-389.
Piccoli, B., Classification of generic singularities for the planar time optimal syntheses. SIAM J. Control Optim. 34 (1996) 914-1946. CrossRef
Piccoli, B. and Sussmann, H.J., Regular synthesis and sufficiency conditions for optimality. SIAM J. Control Optim. 39 (2000) 359-410. CrossRef
Sussmann, H.J., Regular synthesis for time-optimal control of single-input real analytic systems in the plane. SIAM J. Control Optim. 25 (1987) 1145-1162. CrossRef