On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.

Bressan, Aldo. "On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 82.3 (1988): 461-471. <http://eudml.org/doc/289149>.

@article{Bressan1988,
abstract = {In [1] I and II various equivalence theorems are proved; e.g. an ODE $(\mathcal\{E\}) \dot\{z\} = F(t,z,u,\dot\{u\}) \, (\in \mathbb\{R\}^\{m\})$ with a scalar control $u = u(\cdot)$ is linear w.r.t. $\dot\{u\}$ iff $(\alpha)$ its solution $z(u,\cdot)$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta)$$z(u, \cdot)$ satisfies certain conditions by which $1^\{st\}$-order discontinuities of $u$ and $\dot\{u\}$ can be treated satisfactorily. In the case when, for $z = (q, p)$ equation $(\mathcal\{E\})$ is a semi-Hamiltonian system, equivalent to a system of Lagrangian equations of a general type, the importance or compulsory character in many situations, of the conditions hinted at in $(\alpha)$ and $(\beta)$, have received some intuitive justifications in [1] II. In the present paper some of these are replaced by theorems and thus the importance of the above linearity is strengthened. E.g. this linearity is shown, roughly speaking, to follow from the continuity (in the afore-mentioned sense) of the function $u \vdash q ( u , \cdot)$ alone. In the above semi-Hamiltonian case, the linearity of equation $(\mathcal\{E\})$ w.r.t. $u$ is also proved to be invariant under certain transformations of Lagrangian co-ordinates.},
author = {Bressan, Aldo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
keywords = {Lagrangian systems; Feed back theory},
language = {eng},
month = {9},
number = {3},
pages = {461-471},
publisher = {Accademia Nazionale dei Lincei},
title = {On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.},
url = {http://eudml.org/doc/289149},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Bressan, Aldo
TI - On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1988/9//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 3
SP - 461
EP - 471
AB - In [1] I and II various equivalence theorems are proved; e.g. an ODE $(\mathcal{E}) \dot{z} = F(t,z,u,\dot{u}) \, (\in \mathbb{R}^{m})$ with a scalar control $u = u(\cdot)$ is linear w.r.t. $\dot{u}$ iff $(\alpha)$ its solution $z(u,\cdot)$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta)$$z(u, \cdot)$ satisfies certain conditions by which $1^{st}$-order discontinuities of $u$ and $\dot{u}$ can be treated satisfactorily. In the case when, for $z = (q, p)$ equation $(\mathcal{E})$ is a semi-Hamiltonian system, equivalent to a system of Lagrangian equations of a general type, the importance or compulsory character in many situations, of the conditions hinted at in $(\alpha)$ and $(\beta)$, have received some intuitive justifications in [1] II. In the present paper some of these are replaced by theorems and thus the importance of the above linearity is strengthened. E.g. this linearity is shown, roughly speaking, to follow from the continuity (in the afore-mentioned sense) of the function $u \vdash q ( u , \cdot)$ alone. In the above semi-Hamiltonian case, the linearity of equation $(\mathcal{E})$ w.r.t. $u$ is also proved to be invariant under certain transformations of Lagrangian co-ordinates.
LA - eng
KW - Lagrangian systems; Feed back theory
UR - http://eudml.org/doc/289149
ER -