On Lagrangian systems with some coordinates as controls

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 -