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

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 1, pp. 107-118.

Abstract (VO)

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.

Riassunto

Sulla base di risultati ottenuti in [2] o [5] - quali i Lemmi 8.1 e 10.1 - si mostra come applicare ai sistemi Lagrangiani i teoremi generali sulla controllabilità, considerati nella Nota I. Nel caso che $\Sigma$, $\chi$ ed $M$ verifichino le condizioni (11.7) con $\mathcal{Q}$ polinomio nelle $\dot{\gamma}$ si mostra che le condizioni (C) - ossia le (11.8) e le (11.7) con $\mathcal{Q} \equiv 0$ - sono necessarie per poter trattare soddisfacentemente moti iper-impulsivi di $\Sigma$ (in cui anche le posizioni posson subire discontinuità di $1^{a}$ specie). Si fa quanto sopra da un punto di vista generale, riferendosi dapprima ad un sistema matematico $(\mathcal{M}) \dot{z} = F(t, \gamma, z , \dot{\gamma})$ polinomiale in $\dot{\gamma}$ e con $z \in \mathbb{R}^{m}$ e $\gamma \in \mathbb{R}^{M}$. La suddetta trattazione si considera soddisfacente se, ad un generico istante $t$, (i) i valori anteriori $z^{-}$ e $\gamma^{-}$ di $z$ e $\gamma$, e $\gamma^{+}$, determinano $z^{+}$ in un certo modo fisicamente naturale e basato su successioni $\{\gamma_{n} (\cdot)\}$ e $\{z_{n} (\cdot)\}$ di funzioni regolari approssimanti le discontinuità $(\gamma^{-},\gamma^{+})$ e $(z^{-},z^{+})$ di $\gamma(\cdot)$ e $z (\cdot)$, (ii) fissati $z^{-}$ e $\gamma^{-}$, $z^{+}$ risulta funzione continua di $\gamma^{+}$ e (iii) quando $\gamma^{+}$ tende a $\gamma^{-}$, $z (\cdot)$ tende ad una funzione continua e le $z_{n}(\cdot)$ si comportano in un certo modo naturale per certe semplici scelte delle $\gamma_{n}(\cdot)$. Se M > 1, le (i)-(iii) son verificate solo in casi molto eccezionali. Allora si considerano le loro versioni (1-dimensionali) (a)-(c) in cui le (i)-(iii) valgono, per così dire, lungo una traiettoria $\tilde{\gamma}$ ($\in C^{3}$) di $\gamma(\cdot)$ con estremi $\gamma^{-}$ e $\gamma^{+}$ e prefissata ad arbitrio, nel senso che $\tilde{\gamma}$ è la traiettoria di tutte le $\gamma_{n}(\cdot)$, ossia $\gamma_{n}(t) = \tilde{\gamma} \left[ u_{n} (t) \right]$. Inoltre si mostra che una certa versione debole (di una parte) delle condizioni (a)-(c) implica la linearità di $(\mathcal{M})$ in $\dot{\gamma}$. Viceversa questa linearità implica una certa versione forte delle (a)-(c), nel qual caso dico che $(\mathcal{D})$ il parametro ($M$-dimensionale) $\gamma$ di $(\mathcal{M})$ è adatto a subire discontinuità (1 -dimensionali, di $1^{a}$ specie). Riguardo alla terna $(\Sigma,\chi,M)$, v. Sommario in Nota I , per $m = 2N$, $\chi=(q, \gamma)$ e $z=(q,p)$ con $p_{h} = \frac{\partial T}{\partial \dot{q}^{h}}$$(h=1,...,N)$, il sistema differenziale $(\mathcal{M})$ può identificarsi con le equazioni dinamiche di $\Sigma_{\tilde{\gamma}}$ in forma hamiltoniana. Allora la loro linearità rispetto alle $\dot{\gamma}$ risulta condizione necessaria e sufficiente affinché le co-ordinate $\chi$ di $\Sigma$ siano $M$-adatte a (subire) iper-impulsi (1-dimensionali) nel senso che valga $(\mathcal{D})$.

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