Structural discontinuities to approximate some optimization problems with a nonmonotone impulsive character
- Volume: 6, Issue: 2, page 93-109
- ISSN: 1120-6330
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topBressan, Aldo, and Motta, Monica. "Structural discontinuities to approximate some optimization problems with a nonmonotone impulsive character." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6.2 (1995): 93-109. <http://eudml.org/doc/244280>.
@article{Bressan1995,
abstract = {In some preceding works we consider a class \( \mathcal\{OP\} \) of Boltz optimization problems for Lagrangian mechanical systems, where it is relevant a line \( l = l\_\{\gamma(\cdot)\} \), regarded as determined by its (variable) curvature function \( \gamma(\cdot) \) of domain \( \left[ s\_\{0\},s\_\{1\} \right] \). Assume that the problem \( \widetilde\{\mathcal\{P\}\} \in \mathcal\{OP\} \) is regular but has an impulsive monotone character in the sense that near each of some points \( \delta\_\{1\} \) to \( \delta\_\{\nu\} \gamma(\cdot) \) is monotone and \( |\gamma'(\cdot)| \) is very large. In [10] we propose a procedure belonging to the theory of impulsive controls, in order to simplify \( \widetilde\{\mathcal\{P\}\} \) into a structurally discontinuous problem \( \mathcal\{P\} \). This is analogous to treating a biliard ball, disregarding its elasticity properties, as a rigid body bouncing according to a suitable restitution coefficient. Here the afore-mentioned treatment of \( \widetilde\{\mathcal\{P\}\} \) is extended to the case where its impulsive character fails to be monotone. Let \( c\_\{r,0\} \) to \( c\_\{r,m\_\{r\}\} \) be the successive maxima and minima of \( \gamma(\cdot) \) or \( - \gamma(\cdot) \) near \( \delta\_\{r\} (r = 1, \ldots, \nu) \). In constructing the problem \( \mathcal\{P\} \), which simplifies and approximates \( \widetilde\{\mathcal\{P\}\} \) as well as in [10] it is essential to approximate \( l\_\{\gamma(\cdot)\} \) by means of a line \( l\_\{c(\cdot)\} \) with \( c(\cdot) \) discontinuous only at \( \delta\_\{1\}, \ldots, \delta\_\{\nu\} \) and with \( |c'(\cdot)| \) never very large; furthermore now we must take the quantities \( c\_\{r,0\} \) to \( c\_\{r,m\_\{r\}\} \) into account, e.g., by adding a «nonmonotonicity» type at \( \delta\_\{r\} \), which vanishes in the monotone case \( (r = 1, \ldots, \nu) \). Starting from [10] we extend to the afore-mentioned general situation the notions of weak lower limit \( J^\{*\} \) of the functional to minimize, extended admissible process (which has an additional part in each \( \left[ c\_\{r,i-1\},c\_\{r,i\} \right] \)) and extended solution of the problem \( \mathcal\{P\} \), or better \( ( \mathcal\{P\}\_\{\nu\}; \sigma\_\{r,1\}, \ldots , \sigma\_\{r,m\_\{r\}\} ) \) where \( \sigma\_\{r,i\} = c\_\{r,i\} —c\_\{r,i-1\} \)\( (i = 1, \ldots , m\_\{r\}; r = 1, \ldots , \nu) \). In the general case we consider the extended (impulsive) original problem and the extended functional to minimize. This has an impulsive part at each of the points \( \delta\_\{1\} \) to \( \delta\_\{\nu\} \), as well as the differential constraints, complementary equations, and Pontrjagin's optimization conditions. Besides the end conditions at \( s\_\{0\} \) and \( s\_\{1\} \) there are junction conditions at \( \delta\_\{1\} \) to \( \delta\_\{\nu\} \). In the general case being considered we state a version of Pontrjagin's maximum principle and an existence theorem for the extended (impulsive) problem. We also study some properties of \( J^\{*\} \), e.g. when \( J^\{*\} \) is a weak minimum. In particular, within both the monotone case and the nonmonotone one, we show that the quantity \( J^\{*\} \), defined as a certain lower limit, equals the analogous limit; and this is practically a necessary and sufficient condition for the present approximation theory, started in [10], to be satisfactory.},
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; Pontryagin's maximum principle},
language = {eng},
month = {6},
number = {2},
pages = {93-109},
publisher = {Accademia Nazionale dei Lincei},
title = {Structural discontinuities to approximate some optimization problems with a nonmonotone impulsive character},
url = {http://eudml.org/doc/244280},
volume = {6},
year = {1995},
}
TY - JOUR
AU - Bressan, Aldo
AU - Motta, Monica
TI - Structural discontinuities to approximate some optimization problems with a nonmonotone impulsive character
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1995/6//
PB - Accademia Nazionale dei Lincei
VL - 6
IS - 2
SP - 93
EP - 109
AB - In some preceding works we consider a class \( \mathcal{OP} \) of Boltz optimization problems for Lagrangian mechanical systems, where it is relevant a line \( l = l_{\gamma(\cdot)} \), regarded as determined by its (variable) curvature function \( \gamma(\cdot) \) of domain \( \left[ s_{0},s_{1} \right] \). Assume that the problem \( \widetilde{\mathcal{P}} \in \mathcal{OP} \) is regular but has an impulsive monotone character in the sense that near each of some points \( \delta_{1} \) to \( \delta_{\nu} \gamma(\cdot) \) is monotone and \( |\gamma'(\cdot)| \) is very large. In [10] we propose a procedure belonging to the theory of impulsive controls, in order to simplify \( \widetilde{\mathcal{P}} \) into a structurally discontinuous problem \( \mathcal{P} \). This is analogous to treating a biliard ball, disregarding its elasticity properties, as a rigid body bouncing according to a suitable restitution coefficient. Here the afore-mentioned treatment of \( \widetilde{\mathcal{P}} \) is extended to the case where its impulsive character fails to be monotone. Let \( c_{r,0} \) to \( c_{r,m_{r}} \) be the successive maxima and minima of \( \gamma(\cdot) \) or \( - \gamma(\cdot) \) near \( \delta_{r} (r = 1, \ldots, \nu) \). In constructing the problem \( \mathcal{P} \), which simplifies and approximates \( \widetilde{\mathcal{P}} \) as well as in [10] it is essential to approximate \( l_{\gamma(\cdot)} \) by means of a line \( l_{c(\cdot)} \) with \( c(\cdot) \) discontinuous only at \( \delta_{1}, \ldots, \delta_{\nu} \) and with \( |c'(\cdot)| \) never very large; furthermore now we must take the quantities \( c_{r,0} \) to \( c_{r,m_{r}} \) into account, e.g., by adding a «nonmonotonicity» type at \( \delta_{r} \), which vanishes in the monotone case \( (r = 1, \ldots, \nu) \). Starting from [10] we extend to the afore-mentioned general situation the notions of weak lower limit \( J^{*} \) of the functional to minimize, extended admissible process (which has an additional part in each \( \left[ c_{r,i-1},c_{r,i} \right] \)) and extended solution of the problem \( \mathcal{P} \), or better \( ( \mathcal{P}_{\nu}; \sigma_{r,1}, \ldots , \sigma_{r,m_{r}} ) \) where \( \sigma_{r,i} = c_{r,i} —c_{r,i-1} \)\( (i = 1, \ldots , m_{r}; r = 1, \ldots , \nu) \). In the general case we consider the extended (impulsive) original problem and the extended functional to minimize. This has an impulsive part at each of the points \( \delta_{1} \) to \( \delta_{\nu} \), as well as the differential constraints, complementary equations, and Pontrjagin's optimization conditions. Besides the end conditions at \( s_{0} \) and \( s_{1} \) there are junction conditions at \( \delta_{1} \) to \( \delta_{\nu} \). In the general case being considered we state a version of Pontrjagin's maximum principle and an existence theorem for the extended (impulsive) problem. We also study some properties of \( J^{*} \), e.g. when \( J^{*} \) is a weak minimum. In particular, within both the monotone case and the nonmonotone one, we show that the quantity \( J^{*} \), defined as a certain lower limit, equals the analogous limit; and this is practically a necessary and sufficient condition for the present approximation theory, started in [10], to be satisfactory.
LA - eng
KW - Analytical mechanics; Lagrangian systems; Control theory; Pontryagin's maximum principle
UR - http://eudml.org/doc/244280
ER -
References
top- ALBERTO BRESSAN, , On differential systems with impulsive controls. Rend. Sem. Mat. Univ. Padova, 78, 1987, 227-236. MR934514
- 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
- BRESSAN, ALDO, On the application of control theory to certain problems for Lagrangian systems, and hyperimpulsive motions for these. I. On some general mathematical considerations on controllizable parameters. Atti Acc. Lincei Rend. fis., s. 8, vol. 82, fasc. 1, 1988, 91-105. Zbl0669.70029MR999841
- BRESSAN, ALDO, On the application of control theory to certain problems for Lagrangian systems, and hyperimpulsive motions for these. II. Some purely mathematical considerations for hyper-impulsive motions. Atti Acc. Lincei Rend. fis., s. 8, vol. 82, fasc. 1, 1988, 107-118. Zbl0669.70030MR999842
- BRESSAN, ALDO, Hyperimpulsive motions and controllizable coordinates for Lagrangian systems. Atti Acc. Lincei Mem. fis., s. 8, vol. 19, sez. 1, 1990, 197-246. MR1201198
- BRESSAN, ALDO, On some control problems concerning the ski or the swing. Mem. Mat. Acc. Lincei, s. 9, vol. 1, 1991, 149-196. Zbl0744.49017MR1119158
- 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, vol. 2, 1991, 339-343. Zbl0784.70025MR1152638
- 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, vol. 3, 1992, 35-42. Zbl0799.70009MR1159997
- 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, vol. 2, 1993, 5-30. Zbl0816.70019MR1271190
- BRESSAN, ALDO - MOTTA, M., Some optimization problems with a monotone impulsive character. Approximation by means of structural discontinuities. Mem. Mat. Acc. Lincei, s. 9, vol. 2, 1994, 31-52. Zbl0810.93039MR1273635
- BRESSAN, ALDO - MOTTA, M., Some optimization problems for the ski simple because of structural discontinuities. Preprint.
- BRESSAN, ALDO - MOTTA, M., On control problems of minimum time for Lagrangian systems similar to a swing. I. Convexity criteria for sets. Rend. Mat. Acc. Lincei, s. 9, vol. 5, 1994, 247-254. Zbl0814.70019MR1298268
- BRESSAN, ALDO - MOTTA, M., On control problems of minimum time for Lagrangian systems similar to a swing. II. Application of convexity criteria to certain minimum time problems. Rend. Mat. Acc. Lincei, s. 9, vol. 5, 1994, 255-264. Zbl0814.70020MR1298269
- 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
- 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
- PICCOLI, B., Time-optimal control problems for the swing and the ski. International Journal of Control, to appear. Zbl0863.49025MR1637962DOI10.1080/00207179508921606
- RAMPAZZO, F., On Lagrangian systems with some coordinates as controls. Atti Acc. Lincei Rend. fis., s. 8, vol. 82, 1988, 685-695. Zbl0758.70013MR1139816
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