Regular syntheses and solutions to discontinuous ODEs

Alessia Marigo; Benedetto Piccoli

ESAIM: Control, Optimisation and Calculus of Variations (2002)

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

Abstract

top
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

top

Marigo, Alessia, and Piccoli, Benedetto. "Regular syntheses and solutions to discontinuous ODEs." ESAIM: Control, Optimisation and Calculus of Variations 7 (2002): 291-307. <http://eudml.org/doc/245805>.

@article{Marigo2002,
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},
language = {eng},
pages = {291-307},
publisher = {EDP-Sciences},
title = {Regular syntheses and solutions to discontinuous ODEs},
url = {http://eudml.org/doc/245805},
volume = {7},
year = {2002},
}

TY - JOUR
AU - Marigo, Alessia
AU - Piccoli, Benedetto
TI - Regular syntheses and solutions to discontinuous ODEs
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
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
UR - http://eudml.org/doc/245805
ER -

References

top
  1. [1] J.P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag (1984). Zbl0538.34007MR755330
  2. [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. [3] V.G. Boltyanskii, Sufficient conditions for optimality and the justification of the dynamic programming principle. SIAM J. Control Optim. 4 (1966) 326-361. Zbl0143.32004MR197205
  4. [4] U. Boscain and B. Piccoli, Extremal synthesis for generic planar systems. J. Dynam. Control Systems 7 (2001) 209-258. Zbl1013.49025MR1830492
  5. [5] A. Bressan and B. Piccoli, A generic classification of time-optimal planar stabilizing feedbacks. SIAM J. Control Optim. 36 (1998) 12-32. Zbl0910.93044MR1616525
  6. [6] P. Brunovsky, Existence of regular syntheses for general problems. J. Differential Equations 38 (1980) 317-343. Zbl0417.49030MR605053
  7. [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. Zbl0892.93053MR1472857
  8. [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. Zbl0664.34001MR1028776
  9. [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. MR436203
  10. [10] M. Goresky and R. MacPherson, Stratified Morse Theory. Springer Verlag, Berlin (1988). Zbl0639.14012MR932724
  11. [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. Zbl0669.49020MR980227
  12. [12] J.P. Laumond and P. Souères, Shortest paths synthesis for a car-like robot. IEEE Trans. Automat. Control 41 (1996) 672-688. Zbl0864.93076MR1386992
  13. [13] A. Marigo and B. Piccoli, Safety controls and applications to the dubins’ car. Nonlinear Differential Equations and Applications (in print). Zbl1051.49021
  14. [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. [15] R. Murray, Nonlinear control of mechanical systems: A Lagrangian perspective, in IFAC Symposium on Nonlinear Control Systems Design (NOLCOS) (1995) 378-389. 
  16. [16] B. Piccoli, Classification of generic singularities for the planar time optimal syntheses. SIAM J. Control Optim. 34 (1996) 914-1946. Zbl0865.49022MR1416494
  17. [17] B. Piccoli and H.J. Sussmann, Regular synthesis and sufficiency conditions for optimality. SIAM J. Control Optim. 39 (2000) 359-410. Zbl0961.93014MR1788064
  18. [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. Zbl0696.93026MR905037

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.