On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters

Aldo Bressan

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti (1988)

  • Volume: 82, Issue: 1, page 91-105
  • ISSN: 0392-7881

Abstract

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In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control u ( ) . However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control u = u ( ) , where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system Σ , referred to a system χ of Lagrangian co-ordinates, and to try and write some handy conditions, (C), on the coefficients of Σ 's kinetic energy 𝒯 and the Lagrangian components 𝒬 of the forces applied to Σ , at least sufficient to satisfactorily use controls of the second kind. More specifically the conditions (C) sought for, should imply that the last M coordinates in χ are 1-dimensionally controllizable, in the sense that one can satisfactorily treat extremum problems concerning a class Γ γ ~ , Δ , Δ , of controls γ = γ ^ ( t ) = γ ~ [ u ( t ) ] that (i) take as values M -tuples of values of those co-ordinates, (ii) have the same arbitrarily prefixed C 2 -path γ ~ as trajectory, (iii) are Lebesgue integrable in that u ( ) 𝔏 1 ( Δ , Δ ) where Δ and Δ are suitable compact segments of ( Δ ̊ Δ ̊ ) , and (iv) are physically carried out in the above way. One of the aims of [4] is just to write the above conditions (C) by using some recent results in control theory-see [2] where Sussmann's paper [7] is extended from continuous to measurable controls-and some consequences of them presented in [3]. The present work, divided into the Notes I to III, has in part the role of an abstract, in that the works [4] to [5] have not yet been proposed for publication and e.g. conditions (C) are written in Note II, i.e. [6], without proof. In Note I conditions (C) are shown to be necessary for the last M coordinates of χ to be 1-dimensionally controllizable; in doing this proof, this controllizability is regarded to include certain (relatively weak) continuity properties, which are important for checking experimentally the theory being considered, and which (therefore) are analogues of the requirement that the solutions of (physical) differential systems shoud depend on the initial data continuously. Conversely conditions (C) imply that even stronger continuity properties hold for ( Σ , χ , M ) . The above proofs are performed in Note I from the general (purely mathematical) point of view considered in [2], by (also) using some results obtained in ([2] and) [3]. The work [4] also aims at extending the well known theory of impulsive motions, with continuous positions but with velocities suffering first order discontinuities, to a theory of hyper-impulsive motions, in which positions also suffer such discontinuities. In case the components 𝒬 depend on Lagrangian velocities in a certain way, in Note II-see its Summary-one proves some analogues for jumps of the results on controllizability stated in Note I. In Part III some intuitive justifications given in Part II are replaced with theorems; furthermore an invariance property is proved.

How to cite

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Bressan, Aldo. "On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 82.1 (1988): 91-105. <http://eudml.org/doc/289141>.

@article{Bressan1988,
abstract = {In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control $u(\cdot)$. However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control $u=u(\cdot)$, where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system $\Sigma$, referred to a system $\chi$ of Lagrangian co-ordinates, and to try and write some handy conditions, (C), on the coefficients of $\Sigma$'s kinetic energy $\mathcal\{T\}$ and the Lagrangian components $\mathcal\{Q\}_\{\mathcal\{R\}\}$ of the forces applied to $\Sigma$, at least sufficient to satisfactorily use controls of the second kind. More specifically the conditions (C) sought for, should imply that the last M coordinates in $\chi$ are 1-dimensionally controllizable, in the sense that one can satisfactorily treat extremum problems concerning a class $\Gamma_\{\tilde\{\gamma\},\Delta,\Delta^\{\prime\}\}$, of controls $\gamma = \hat\{\gamma\}(t) = \tilde\{\gamma\} \left[u(t)\right]$ that (i) take as values $M$-tuples of values of those co-ordinates, (ii) have the same arbitrarily prefixed $C^\{2\}$-path $\tilde\{\gamma\}$ as trajectory, (iii) are Lebesgue integrable in that $u(\cdot) \in \mathfrak\{L\}^\{1\} (\Delta,\Delta^\{\prime\})$ where $\Delta$ and $\Delta^\{\prime\}$ are suitable compact segments of $\mathbb\{R\}(\mathring\{\Delta\} \ne \emptyset \ne \mathring\{\Delta\}^\{\prime\})$, and (iv) are physically carried out in the above way. One of the aims of [4] is just to write the above conditions (C) by using some recent results in control theory-see [2] where Sussmann's paper [7] is extended from continuous to measurable controls-and some consequences of them presented in [3]. The present work, divided into the Notes I to III, has in part the role of an abstract, in that the works [4] to [5] have not yet been proposed for publication and e.g. conditions (C) are written in Note II, i.e. [6], without proof. In Note I conditions (C) are shown to be necessary for the last $M$ coordinates of $\chi$ to be 1-dimensionally controllizable; in doing this proof, this controllizability is regarded to include certain (relatively weak) continuity properties, which are important for checking experimentally the theory being considered, and which (therefore) are analogues of the requirement that the solutions of (physical) differential systems shoud depend on the initial data continuously. Conversely conditions (C) imply that even stronger continuity properties hold for $(\Sigma,\chi,M)$. The above proofs are performed in Note I from the general (purely mathematical) point of view considered in [2], by (also) using some results obtained in ([2] and) [3]. The work [4] also aims at extending the well known theory of impulsive motions, with continuous positions but with velocities suffering first order discontinuities, to a theory of hyper-impulsive motions, in which positions also suffer such discontinuities. In case the components $\mathcal\{Q\}_\{\mathcal\{R\}\}$ depend on Lagrangian velocities in a certain way, in Note II-see its Summary-one proves some analogues for jumps of the results on controllizability stated in Note I. In Part III some intuitive justifications given in Part II are replaced with theorems; furthermore an invariance property is proved.},
author = {Bressan, Aldo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
keywords = {Mathematical-physics; Feedback theory},
language = {eng},
month = {3},
number = {1},
pages = {91-105},
publisher = {Accademia Nazionale dei Lincei},
title = {On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters},
url = {http://eudml.org/doc/289141},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Bressan, Aldo
TI - On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1988/3//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 1
SP - 91
EP - 105
AB - In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control $u(\cdot)$. However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control $u=u(\cdot)$, where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system $\Sigma$, referred to a system $\chi$ of Lagrangian co-ordinates, and to try and write some handy conditions, (C), on the coefficients of $\Sigma$'s kinetic energy $\mathcal{T}$ and the Lagrangian components $\mathcal{Q}_{\mathcal{R}}$ of the forces applied to $\Sigma$, at least sufficient to satisfactorily use controls of the second kind. More specifically the conditions (C) sought for, should imply that the last M coordinates in $\chi$ are 1-dimensionally controllizable, in the sense that one can satisfactorily treat extremum problems concerning a class $\Gamma_{\tilde{\gamma},\Delta,\Delta^{\prime}}$, of controls $\gamma = \hat{\gamma}(t) = \tilde{\gamma} \left[u(t)\right]$ that (i) take as values $M$-tuples of values of those co-ordinates, (ii) have the same arbitrarily prefixed $C^{2}$-path $\tilde{\gamma}$ as trajectory, (iii) are Lebesgue integrable in that $u(\cdot) \in \mathfrak{L}^{1} (\Delta,\Delta^{\prime})$ where $\Delta$ and $\Delta^{\prime}$ are suitable compact segments of $\mathbb{R}(\mathring{\Delta} \ne \emptyset \ne \mathring{\Delta}^{\prime})$, and (iv) are physically carried out in the above way. One of the aims of [4] is just to write the above conditions (C) by using some recent results in control theory-see [2] where Sussmann's paper [7] is extended from continuous to measurable controls-and some consequences of them presented in [3]. The present work, divided into the Notes I to III, has in part the role of an abstract, in that the works [4] to [5] have not yet been proposed for publication and e.g. conditions (C) are written in Note II, i.e. [6], without proof. In Note I conditions (C) are shown to be necessary for the last $M$ coordinates of $\chi$ to be 1-dimensionally controllizable; in doing this proof, this controllizability is regarded to include certain (relatively weak) continuity properties, which are important for checking experimentally the theory being considered, and which (therefore) are analogues of the requirement that the solutions of (physical) differential systems shoud depend on the initial data continuously. Conversely conditions (C) imply that even stronger continuity properties hold for $(\Sigma,\chi,M)$. The above proofs are performed in Note I from the general (purely mathematical) point of view considered in [2], by (also) using some results obtained in ([2] and) [3]. The work [4] also aims at extending the well known theory of impulsive motions, with continuous positions but with velocities suffering first order discontinuities, to a theory of hyper-impulsive motions, in which positions also suffer such discontinuities. In case the components $\mathcal{Q}_{\mathcal{R}}$ depend on Lagrangian velocities in a certain way, in Note II-see its Summary-one proves some analogues for jumps of the results on controllizability stated in Note I. In Part III some intuitive justifications given in Part II are replaced with theorems; furthermore an invariance property is proved.
LA - eng
KW - Mathematical-physics; Feedback theory
UR - http://eudml.org/doc/289141
ER -

References

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  1. ALEXÉEV, V., TIKHOMIROV, V. and FOMINE, G. (1982) - Commande optimale, Mir, Moscow, 447 pp. MR728225
  2. BRESSAN, ALBERTO (1987) - On differential systems with impulsive controls, «Sem. Mat. Univ. Padova», 78, p. 227. MR934514
  3. BRESSAN, ALDO - On some recent results in control theory, for their application to Lagrangian systems (memoir being printed on «Atti Accad. dei Lincei»), 
  4. BRESSAN, ALDO - Hyper-impulsive motions and controllizable co-ordinates for Lagrangian systems (in preparation). 
  5. BRESSAN, ALDO - On some control problems concerning the ski or swing (in preparation). Zbl0744.49017
  6. BRESSAN, ALDO - Some purely mathematical considerations for hyperimpulsive motions. Applications to Lagrangian systems (being Printed on Atti Accad. dei Lincei). Zbl0669.70030
  7. SUSSMANN, H.J. (1978) - On the gap between deterministic and stocastic ordinary differential equations, «Ann. of Probability», 6 (1978), pp. 19-41. Zbl0391.60056MR461664

Citations in EuDML Documents

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  1. Aldo Bressan, On control theory and its applications to certain problems for Lagrangian systems. On hyperimpulsive motions for these. II. Some purely mathematical considerations for hyper-impulsive motions. Applications to Lagrangian systems
  2. Aldo Bressan, Marco Favretti, 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
  3. Aldo Bressan, Monica Motta, On control problems of minimum time for Lagrangian systems similar to a swing. I. Convexity criteria for sets
  4. Aldo Bressan, Monica Motta, Structural discontinuities to approximate some optimization problems with a nonmonotone impulsive character
  5. Aldo Bressan, On the alpine ski with dry friction and air resistance. Some optimization problems for it

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