Regular syntheses and solutions to discontinuous ODEs

Alessia Marigo; Benedetto Piccoli

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 7, page 291-307
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Marigo, Alessia, and Piccoli, Benedetto. "Regular syntheses and solutions to discontinuous ODEs." ESAIM: Control, Optimisation and Calculus of Variations 7 (2010): 291-307. <http://eudml.org/doc/90624>.

@article{Marigo2010,
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. },
author = {Marigo, Alessia, Piccoli, Benedetto},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Optimal control; regular synthesis; discontinuous ODEs.; optimal control; discontinuous ODEs},
language = {eng},
month = {3},
pages = {291-307},
publisher = {EDP Sciences},
title = {Regular syntheses and solutions to discontinuous ODEs},
url = {http://eudml.org/doc/90624},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Marigo, Alessia
AU - Piccoli, Benedetto
TI - Regular syntheses and solutions to discontinuous ODEs
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 291
EP - 307
AB - 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.
LA - eng
KW - Optimal control; regular synthesis; discontinuous ODEs.; optimal control; discontinuous ODEs
UR - http://eudml.org/doc/90624
ER -

References

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  7. F.H. Clarke, Y.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control42 (1997) 1394-1407.  Zbl0892.93053
  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.  
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  10. M. Goresky and R. MacPherson, Stratified Morse Theory. Springer Verlag, Berlin (1988).  
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  12. J.P. Laumond and P. Souères, Shortest paths synthesis for a car-like robot. IEEE Trans. Automat. Control41 (1996) 672-688.  Zbl0864.93076
  13. A. Marigo and B. Piccoli, Safety controls and applications to the dubins' car. Nonlinear Differential Equations and Applications (in print).  Zbl1051.49021
  14. A. Marigo and B. Piccoli, Safety driving for the dubins' car, in XV World Congress on Automatic Control b'02 (in print).  Zbl1051.49021
  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.  Zbl0865.49022
  17. B. Piccoli and H.J. Sussmann, Regular synthesis and sufficiency conditions for optimality. SIAM J. Control Optim.39 (2000) 359-410.  Zbl0961.93014
  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.  Zbl0701.93035

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