On control problems of minimum time for Lagrangian systems similar to a swing. I. Convexity criteria for sets

Aldo Bressan; Monica Motta

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1994)

  • Volume: 5, Issue: 3, page 247-254
  • ISSN: 1120-6330

Abstract

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One establishes some convexity criteria for sets in R 2 . They will be applied in a further Note to treat the existence of solutions to minimum time problems for certain Lagrangian systems referred to two coordinates, one of which is used as a control. These problems regard the swing or the ski.

How to cite

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Bressan, Aldo, and Motta, Monica. "On control problems of minimum time for Lagrangian systems similar to a swing. I. Convexity criteria for sets." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 5.3 (1994): 247-254. <http://eudml.org/doc/244283>.

@article{Bressan1994,
abstract = {One establishes some convexity criteria for sets in \( R^\{2\} \). They will be applied in a further Note to treat the existence of solutions to minimum time problems for certain Lagrangian systems referred to two coordinates, one of which is used as a control. These problems regard the swing or the ski.},
author = {Bressan, Aldo, Motta, Monica},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Analytical mechanics; Lagrangian systems; Control theory; existence of solutions; ski},
language = {eng},
month = {9},
number = {3},
pages = {247-254},
publisher = {Accademia Nazionale dei Lincei},
title = {On control problems of minimum time for Lagrangian systems similar to a swing. I. Convexity criteria for sets},
url = {http://eudml.org/doc/244283},
volume = {5},
year = {1994},
}

TY - JOUR
AU - Bressan, Aldo
AU - Motta, Monica
TI - On control problems of minimum time for Lagrangian systems similar to a swing. I. Convexity criteria for sets
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1994/9//
PB - Accademia Nazionale dei Lincei
VL - 5
IS - 3
SP - 247
EP - 254
AB - One establishes some convexity criteria for sets in \( R^{2} \). They will be applied in a further Note to treat the existence of solutions to minimum time problems for certain Lagrangian systems referred to two coordinates, one of which is used as a control. These problems regard the swing or the ski.
LA - eng
KW - Analytical mechanics; Lagrangian systems; Control theory; existence of solutions; ski
UR - http://eudml.org/doc/244283
ER -

References

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  1. BRESSAN, ALBERTO, On differential systems with impulsive controls. Rend. Sem. Mat. Univ. Padova, 78, 1987, 227-236. MR934514
  2. BRESSAN, ALBERTO - RAMPAZZO, F., Impulsive control systems with commutative vector fields. Journal of optimization theory and applications, 71, 1991, 67-83. Zbl0793.49014MR1131450DOI10.1007/BF00940040
  3. BRESSAN, ALDO, On the application of control theory to certain problems for Lagrangian systems, and hyperimpulsive motion for these. I. Some general mathematical considerations on controllizable parameters. Atti Acc. Lincei Rend, fis., s. 8, v. 82, 1988, 91-105. Zbl0669.70029MR999841
  4. BRESSAN, ALDO, On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. II. Some purely mathematical considerations for hyper-impulsive motions. Applications to Lagrangian systems. Atti Acc. Lincei Rend, fis., s. 8, v. 82, 1988, 107-118. Zbl0669.70030MR999842
  5. BRESSAN, ALDO, Hyper-impulsive motions and controllizable coordinates for Lagrangian systems. Atti Acc. Lincei Mem. fis., s. 8, v. 19, sez. I, fasc. 7 (1989), 1990, 195-246. MR1163634
  6. BRESSAN, ALDO, On some control problems concerning the ski or the swing. Mem. Mat. Acc. Lincei, s. 9, v. 1, 1991, 147-196. Zbl0744.49017MR1119158
  7. BRESSAN, ALDO - FAVRETTI, M., On motions with bursting characters for Lagrangian mechanical systems with a scalar control. I. Existence of a wide class of Lagrangian systems capable of motions with bursting characters. Rend. Mat. Acc. Lincei, s. 9, v. 2, 1991, 339-343. Zbl0784.70025MR1152638
  8. BRESSAN, ALDO - FAVRETTI, M., On motions with bursting characters for Lagrangian mechanical systems with a scalar control. II. A geodesic property of motions with bursting characters for Lagrangian systems. Rend. Mat. Acc. Lincei, s. 9, v. 3, 1992, 35-42. Zbl0799.70009MR1159997
  9. BRESSAN, ALDO - MOTTA, M., A class of mechanical systems with some coordinates as controls. A reduction of certain optimization problems for them. Solutions methods. Mem. Mat. Acc. Lincei, s. 9, v. 2, 1993, 5-30. Zbl0816.70019MR1271190
  10. BRESSAN, ALDO - MOTTA, M., Some optimization problems with a monotone impulsive character. Approximation by means of structural discontinuities. Mem. Mat. Acc. Lincei, s. 9, v. 2, 1994, 31-52. Zbl0810.93039MR1273635
  11. BRESSAN, ALDO - MOTTA, M., Some optimization problems for the ski simple because of structural discontinuities. Preprint. 
  12. BRESSAN, ALDO - MOTTA, M., Structural discontinuities to approximate some optimization problems with a nonmonotone impulsive character. Preprint. Zbl0858.70012MR1354223
  13. BRESSAN, ALDO - MOTTA, M., On minimum time problems for a pendulum with variable length and a conjecture based on a law of Galilei. Atti Istituto Veneto di Scienze Lettere ed Arti. To appear. Zbl0882.70021MR1360156
  14. FAVRETTI, M., Essential character of the assumptions of a theorem of Aldo Bressan on the coordinates of a Lagrangian system that are fit for jumps. Atti Istituto Veneto di Scienze Lettere ed Arti, 149, 1991, 1-14. Zbl0783.70020MR1237965
  15. FAVRETTI, M., Some bounds for the solutions of certain families of Cauchy problems connected with bursting phenomena. Atti Istituto Veneto di Scienze Lettere ed Arti, 149, 1991, 61-75. Zbl0795.70019MR1237966
  16. LEE, E. B. - MARKUS, L., Foundations of Optimal Control Theory. SIAMseries in Applied Mathematics, John Wiley & Sons, Inc., New York-London-Sidney 1977. Zbl0159.13201MR220537
  17. RAMPAZZO, F., On Lagrangian systems with some coordinates as controls. Atti Acc. Lincei Rend, fis., s. 8, v. 82, 1988, 685-695. Zbl0758.70013MR1139816

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