On Lagrangian systems with some coordinates as controls

Franco Rampazzo

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

  • Volume: 82, Issue: 4, page 685-695
  • ISSN: 0392-7881

Abstract

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Let Σ be a constrained mechanical system locally referred to state coordinates ( q 1 , , q N , γ 1 , , γ M ) . Let ( γ ~ 1 γ ~ M ) ( ) be an assigned trajectory for the coordinates γ α and let u ( ) be a scalar function of the time, to be thought as a control. In [4] one considers the control system Σ γ ^ , which is parametrized by the coordinates ( q 1 , , q N ) and is obtained from Σ by adding the time-dependent, holonomic constraints γ α = γ ^ α ( t ) := γ ~ α ( u ( t ) ) . More generally, one can consider a vector-valued control u ( ) = ( u 1 , , u M ) ( ) which is directly identified with γ ^ ( ) = ( γ ^ 1 , , γ ^ M ) ( ) . If one denotes the momenta conjugate to the coordinates q i by p i , i = 1 , , N , it is physically interesting to examine the continuity properties of the input-output map ϕ : u ( ) ( q i , p i ) ( ) associated with the dynamical equations of Σ γ ^ with respect to e.g. the C 0 topologies on the spaces of the controls u ( ) and of the solutions ( q i , p i ) ( ) . Furthermore, in the theory of hyperimpulsive motions (see [4]), even discontinuous control are implemented. Then it is crucial to investigate the continuity of ϕ also with respect to topologies that are weaker than the C 0 one. In order that the input-output map ϕ exhibits such continuity properties, the right-hand sides of the dynamical equation for Σ γ ^ have to be affine in the derivatives d γ ^ 1 d t , , d γ ^ M d t . If this is the case, the system of coordinates ( q i , γ α ) is said to be M -fit (for linearity). In this note we show that, in the case of forces which depend linearly on the velocity of Σ , the coordinate system q i , γ α ) is M -fit if and only if certain coefficients in the expression of the kinetic energy are independent of the q i . Moreover, if the forces are positional, for each 1 -fit coordinate system ( q i , y ) there exists a reparametrization ( q j , γ ) such that γ q i = 0 holds for every i = 1 , , N and the coordinates ( q i , γ ) are locally geodesic.

How to cite

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Rampazzo, Franco. "On Lagrangian systems with some coordinates as controls." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 82.4 (1988): 685-695. <http://eudml.org/doc/289275>.

@article{Rampazzo1988,
abstract = {Let $\Sigma$ be a constrained mechanical system locally referred to state coordinates $(q^\{1\},...,q^\{N\}, \gamma^\{1\},...,\gamma^\{M\})$. Let $(\tilde\{\gamma\}^\{1\}...\tilde\{\gamma\}^\{M\})(\cdot)$ be an assigned trajectory for the coordinates $\gamma^\{\alpha\}$ and let $u(\cdot)$ be a scalar function of the time, to be thought as a control. In [4] one considers the control system $\Sigma_\{\hat\{\gamma\}\}$, which is parametrized by the coordinates $(q^\{1\},...,q^\{N\})$ and is obtained from $\Sigma$ by adding the time-dependent, holonomic constraints $\gamma^\{\alpha\} = \hat\{\gamma\}^\{\alpha\}(t) := \tilde\{\gamma\}^\{\alpha\} (u(t))$. More generally, one can consider a vector-valued control $u(\cdot) = (u^\{1\},..., u^\{M\})(\cdot)$ which is directly identified with $\hat\{\gamma\}(\cdot) = (\hat\{\gamma\}^\{1\},..., \hat\{\gamma\}^\{M\})(\cdot)$. If one denotes the momenta conjugate to the coordinates $q^\{i\}$ by $p_\{i\}$, $i= 1,...,N$, it is physically interesting to examine the continuity properties of the input-output map $\phi : u(\cdot) \rightarrow (q^\{i\} ,p_\{i\})(\cdot)$ associated with the dynamical equations of $\Sigma_\{\hat\{\gamma\}\}$ with respect to e.g. the $C^\{0\}$ topologies on the spaces of the controls $u(\cdot)$ and of the solutions $(q^\{i\},p_\{i\})(\cdot)$. Furthermore, in the theory of hyperimpulsive motions (see [4]), even discontinuous control are implemented. Then it is crucial to investigate the continuity of $\phi$ also with respect to topologies that are weaker than the $C^\{0\}$ one. In order that the input-output map $\phi$ exhibits such continuity properties, the right-hand sides of the dynamical equation for $\Sigma_\{\hat\{\gamma\}\}$ have to be affine in the derivatives $\frac\{d \hat\{\gamma\}^\{1\}\} \{dt\},...,\frac\{d\hat\{\gamma\}^\{M\}\} \{dt\}$. If this is the case, the system of coordinates $(q^\{i\} ,\gamma^\{\alpha\})$ is said to be $M$-fit (for linearity). In this note we show that, in the case of forces which depend linearly on the velocity of $\Sigma$, the coordinate system $q^\{i\} ,\gamma^\{\alpha\})$ is $M$-fit if and only if certain coefficients in the expression of the kinetic energy are independent of the $q^\{i\}$. Moreover, if the forces are positional, for each $1$-fit coordinate system $(q^\{\prime i\} ,y^\{\prime\})$ there exists a reparametrization $(q^\{j\} ,\gamma)$ such that $\frac\{\partial\gamma\}\{\partial q^\{\prime i\}\} = 0$ holds for every $i = 1,...,N$ and the coordinates $(q^\{i\} ,\gamma)$ are locally geodesic.},
author = {Rampazzo, Franco},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
keywords = {Lagrangian systems; Impulsive controls; Kinetic metric},
language = {eng},
month = {12},
number = {4},
pages = {685-695},
publisher = {Accademia Nazionale dei Lincei},
title = {On Lagrangian systems with some coordinates as controls},
url = {http://eudml.org/doc/289275},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Rampazzo, Franco
TI - On Lagrangian systems with some coordinates as controls
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1988/12//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 4
SP - 685
EP - 695
AB - Let $\Sigma$ be a constrained mechanical system locally referred to state coordinates $(q^{1},...,q^{N}, \gamma^{1},...,\gamma^{M})$. Let $(\tilde{\gamma}^{1}...\tilde{\gamma}^{M})(\cdot)$ be an assigned trajectory for the coordinates $\gamma^{\alpha}$ and let $u(\cdot)$ be a scalar function of the time, to be thought as a control. In [4] one considers the control system $\Sigma_{\hat{\gamma}}$, which is parametrized by the coordinates $(q^{1},...,q^{N})$ and is obtained from $\Sigma$ by adding the time-dependent, holonomic constraints $\gamma^{\alpha} = \hat{\gamma}^{\alpha}(t) := \tilde{\gamma}^{\alpha} (u(t))$. More generally, one can consider a vector-valued control $u(\cdot) = (u^{1},..., u^{M})(\cdot)$ which is directly identified with $\hat{\gamma}(\cdot) = (\hat{\gamma}^{1},..., \hat{\gamma}^{M})(\cdot)$. If one denotes the momenta conjugate to the coordinates $q^{i}$ by $p_{i}$, $i= 1,...,N$, it is physically interesting to examine the continuity properties of the input-output map $\phi : u(\cdot) \rightarrow (q^{i} ,p_{i})(\cdot)$ associated with the dynamical equations of $\Sigma_{\hat{\gamma}}$ with respect to e.g. the $C^{0}$ topologies on the spaces of the controls $u(\cdot)$ and of the solutions $(q^{i},p_{i})(\cdot)$. Furthermore, in the theory of hyperimpulsive motions (see [4]), even discontinuous control are implemented. Then it is crucial to investigate the continuity of $\phi$ also with respect to topologies that are weaker than the $C^{0}$ one. In order that the input-output map $\phi$ exhibits such continuity properties, the right-hand sides of the dynamical equation for $\Sigma_{\hat{\gamma}}$ have to be affine in the derivatives $\frac{d \hat{\gamma}^{1}} {dt},...,\frac{d\hat{\gamma}^{M}} {dt}$. If this is the case, the system of coordinates $(q^{i} ,\gamma^{\alpha})$ is said to be $M$-fit (for linearity). In this note we show that, in the case of forces which depend linearly on the velocity of $\Sigma$, the coordinate system $q^{i} ,\gamma^{\alpha})$ is $M$-fit if and only if certain coefficients in the expression of the kinetic energy are independent of the $q^{i}$. Moreover, if the forces are positional, for each $1$-fit coordinate system $(q^{\prime i} ,y^{\prime})$ there exists a reparametrization $(q^{j} ,\gamma)$ such that $\frac{\partial\gamma}{\partial q^{\prime i}} = 0$ holds for every $i = 1,...,N$ and the coordinates $(q^{i} ,\gamma)$ are locally geodesic.
LA - eng
KW - Lagrangian systems; Impulsive controls; Kinetic metric
UR - http://eudml.org/doc/289275
ER -

References

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  1. ARNOLD, V. (1980) - Equazioni differenziali ordinarie, MIR, Moscow. 
  2. BRESSAN, A. (1987) - On differential systems with impulsive controls, Rend. Sem.. Mat. Un. Padova, 78, 227-236. MR934514
  3. BRESSAN, A. and RAMPAZZO, F. (1988) - On differential systems with vector-valued impulsive control, Boll. Un. Mat. Ital. B, 3, 641-656. Zbl0653.49002MR963323
  4. BRESSAN, A. (1990) - Hyperimpulsive motions and controllizahle coordinates for Lagrangian systems, To appear on Atti Accad. Naz. Lincei, Rend. Cl. Sc. Fis. Mat. Natur. 
  5. BRESSAN, A. (1990) - On some control problems concerning the ski and the swing, To appear. MR1119158
  6. BRESSAN, A. (1990) - On some recent results in control theory, for their applications to Lagrangian systems, To appear on Atti Accad. Naz. Lincei, Rend. Cl. Sc. Fis. Mat. Natur. MR1201198
  7. BRESSAN, A. (1988) - On the applications of control theory to certain problems for Lagrangian systems, and hyperimpulsive motions for these. I and II, Atti Accad. Naz. Lincei, Rend. Cl. Sc. Fis. Mat. Nat.82-1, 91-105 and 107-118. Zbl0669.70030
  8. SUSSMANN, H.J. (1978) - On the gap between deterministic and stochastic ordinary differential equations, Ann. of probability, 6(1978), 19-41. Zbl0391.60056MR461664

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