# 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

- Volume: 82, Issue: 1, page 107-118
- ISSN: 0392-7881

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topBressan, 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 82.1 (1988): 107-118. <http://eudml.org/doc/289024>.

@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},

keywords = {Mathematical physics; Feed-back theory},

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/289024},

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

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

UR - http://eudml.org/doc/289024

ER -

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## Citations in EuDML Documents

top- Franco Rampazzo, On Lagrangian systems with some coordinates as controls
- 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
- Aldo Bressan, Monica Motta, On control problems of minimum time for Lagrangian systems similar to a swing. I. Convexity criteria for sets
- Aldo Bressan, Monica Motta, Structural discontinuities to approximate some optimization problems with a nonmonotone impulsive character
- Aldo Bressan, On the alpine ski with dry friction and air resistance. Some optimization problems for it

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