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

Bressan, Aldo. "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." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 82.1 (1988): 107-118. <http://eudml.org/doc/287395>.

@article{Bressan1988,
abstract = {See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case $\Sigma$, $\chi$ and $M$ satisfy conditions (11.7) when $\mathcal\{Q\}$ is a polynomial in $\dot\{\gamma\}$, conditions (C)-i.e. (11.8) and (11.7) with $\mathcal\{Q\} \equiv 0$-are proved to be necessary for treating satisfactorily $\Sigma$'s hyper-impulsive motions (in which positions can suffer first order discontinuities). This is done from a general point of view, by referring to a mathematical system $(\mathcal\{M\}) \dot\{z\} = F(t, \gamma, z , \dot\{\gamma\})$ where $z \in \mathbb\{R\}^\{m\}$, $\gamma \in \mathbb\{R\}^\{M\}$, and $F(\cdots)$ is a polynomial in $\dot\{\gamma\}$. The afore-mentioned treatment is considered satisfactory when, at a typical instant $t$, (i) the anterior values $z^\{-\}$ and $\gamma^\{-\}$ of $z$ and $\gamma$, together with $\gamma^\{+\}$ determine $z^\{+\}$ in a certain physically natural way, based on certain sequences $\\{ \gamma_\{n\}(\cdot)\\}$ and $\\{ z_\{n\}(\cdot)\\}$ of regular functions that approximate the 1st order discontinuities $(\gamma^\{-\},\gamma^\{+\})$ and $(z^\{-\},z^\{+\})$ of $\gamma(\cdot)$ and $z(\cdot)$ respectively, (ii) for $z^\{-\}$ and $\gamma^\{-\}$ fixed, $z^\{+\}$ is a continuous function of $\gamma^\{+\}$, and (iii) if $\gamma^\{+\}$ tends to $\gamma^\{-\}$, then $z(\cdot)$ tends to a continuous function and, for certain simple choices of $\\{ \gamma_\{n\}(\cdot) \\}$ the functions $z_\{n\}(\cdot)$ behave in a certain natural way. For M > 1, conditions (i) to (iii) hold only in very exceptional cases. Then their 1-dimensional versions (a) to (c) are considered, according to which (i) to (iii) hold, so to say, along a trajectory $\tilde\{\gamma\}$$(\in C^\{3\})$ of $\gamma$'s discontinuity $(\gamma^\{-\},\gamma^\{+\})$, chosen arbitrarily; this means that $\tilde\{\gamma\}$ belongs to the trajectory of all regular functions $\gamma_\{n\}(\cdot)$$(n \in \mathbf\{N\})$, i.e. $\gamma_\{n\}(t) \equiv \tilde\{\gamma\} \left[ u_\{n\}(t) \right]$. Furthermore a certain weak version of (a part of) conditions (a) to (c) is proved to imply the linearity of $(\mathcal\{M\})$. Conversely this linearity implies a strong version of conditions (a) to (c); and when this version holds, one can say that $(\mathcal\{D\})$ the ($M$-dimensional) parameter $\gamma$ in $(\mathcal\{M\})$ is fit to suffer (1-dimensional first order) discontinuities. As far as the triplet $(\Sigma,\chi,M)$, see Summary in Nota I, is concerned, for $m = 2N$, $\chi = (q,\gamma)$, and $z = (q,p)$ with $p_\{h\} = \frac\{\partial T\}\{\partial\dot\{q\}_\{h\}\}$$(h=1,...,N)$, the differential system $(\mathcal\{M\})$ can be identified with the dynamic equations of $\Sigma_\{\hat\{\gamma\}\}$ in semi-hamiltonian form. Then its linearity in $\dot\{\gamma\}$ is necessary and sufficient for the co-ordinates $\chi$ of $\Sigma$ to be $M$-fit for (1-dimensional) hyper-impulses, in the sense that $(\mathcal\{D\})$ holds.},
author = {Bressan, Aldo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Mathematical physics; Feed-back theory; Lagrangian systems; hyper-impulsive motions},
language = {eng},
month = {3},
number = {1},
pages = {107-118},
publisher = {Accademia Nazionale dei Lincei},
title = {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},
url = {http://eudml.org/doc/287395},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Bressan, Aldo
TI - 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
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1988/3//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 1
SP - 107
EP - 118
AB - See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case $\Sigma$, $\chi$ and $M$ satisfy conditions (11.7) when $\mathcal{Q}$ is a polynomial in $\dot{\gamma}$, conditions (C)-i.e. (11.8) and (11.7) with $\mathcal{Q} \equiv 0$-are proved to be necessary for treating satisfactorily $\Sigma$'s hyper-impulsive motions (in which positions can suffer first order discontinuities). This is done from a general point of view, by referring to a mathematical system $(\mathcal{M}) \dot{z} = F(t, \gamma, z , \dot{\gamma})$ where $z \in \mathbb{R}^{m}$, $\gamma \in \mathbb{R}^{M}$, and $F(\cdots)$ is a polynomial in $\dot{\gamma}$. The afore-mentioned treatment is considered satisfactory when, at a typical instant $t$, (i) the anterior values $z^{-}$ and $\gamma^{-}$ of $z$ and $\gamma$, together with $\gamma^{+}$ determine $z^{+}$ in a certain physically natural way, based on certain sequences $\{ \gamma_{n}(\cdot)\}$ and $\{ z_{n}(\cdot)\}$ of regular functions that approximate the 1st order discontinuities $(\gamma^{-},\gamma^{+})$ and $(z^{-},z^{+})$ of $\gamma(\cdot)$ and $z(\cdot)$ respectively, (ii) for $z^{-}$ and $\gamma^{-}$ fixed, $z^{+}$ is a continuous function of $\gamma^{+}$, and (iii) if $\gamma^{+}$ tends to $\gamma^{-}$, then $z(\cdot)$ tends to a continuous function and, for certain simple choices of $\{ \gamma_{n}(\cdot) \}$ the functions $z_{n}(\cdot)$ behave in a certain natural way. For M > 1, conditions (i) to (iii) hold only in very exceptional cases. Then their 1-dimensional versions (a) to (c) are considered, according to which (i) to (iii) hold, so to say, along a trajectory $\tilde{\gamma}$$(\in C^{3})$ of $\gamma$'s discontinuity $(\gamma^{-},\gamma^{+})$, chosen arbitrarily; this means that $\tilde{\gamma}$ belongs to the trajectory of all regular functions $\gamma_{n}(\cdot)$$(n \in \mathbf{N})$, i.e. $\gamma_{n}(t) \equiv \tilde{\gamma} \left[ u_{n}(t) \right]$. Furthermore a certain weak version of (a part of) conditions (a) to (c) is proved to imply the linearity of $(\mathcal{M})$. Conversely this linearity implies a strong version of conditions (a) to (c); and when this version holds, one can say that $(\mathcal{D})$ the ($M$-dimensional) parameter $\gamma$ in $(\mathcal{M})$ is fit to suffer (1-dimensional first order) discontinuities. As far as the triplet $(\Sigma,\chi,M)$, see Summary in Nota I, is concerned, for $m = 2N$, $\chi = (q,\gamma)$, and $z = (q,p)$ with $p_{h} = \frac{\partial T}{\partial\dot{q}_{h}}$$(h=1,...,N)$, the differential system $(\mathcal{M})$ can be identified with the dynamic equations of $\Sigma_{\hat{\gamma}}$ in semi-hamiltonian form. Then its linearity in $\dot{\gamma}$ is necessary and sufficient for the co-ordinates $\chi$ of $\Sigma$ to be $M$-fit for (1-dimensional) hyper-impulses, in the sense that $(\mathcal{D})$ holds.
LA - eng
KW - Mathematical physics; Feed-back theory; Lagrangian systems; hyper-impulsive motions
UR - http://eudml.org/doc/287395
ER -