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

Bressan, Aldo; Favretti, Marco

Bressan, Aldo ; Favretti, Marco

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 2 (1991), p. 339-343 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
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In this Note (which will be followed by a second) we consider a Lagrangian system $Σ$ (possibly without any Lagrangian function) referred to $N + 1$ coordinates $q_{1} \dots, q_{N}$ , $u$ , with $u$ to be used as a control, and precisely to add to $Σ$ a frictionless constraint of the type $u = u (t)$ . Let $Σ$ 's (frictionless) constraints be represented by the manifold $V_{t}$ generally moving in Hertz's space. We also consider an instant $d$ (to be used for certain limit discontinuity-properties), a point $(\bar{q}, \bar{u})$ of $V_{d}$ , a value $\bar{p}$ for $Σ$ 's momentum conjugate to $q$ , and a continuous control $v (\cdot)$ with $v (d) = \bar{u}$ . Furthermore zero is assumed not to equal a certain quantity determined by $Σ$ 's kinetic energy and $Σ$ 's applied forces, which forces are assumed to be at most linear in $\dot{u}$ . A purely mathematical work of Favretti allows us to quickly show that (i) $v (\cdot)$ is the $C^{0}$ -limit of a sequence $u_{a} (\cdot)$ of continuous controls that have a jump character in some interval $[d, d + η_{a}]$ and satisfy certain conditions including that both $η_{a} \to 0^{+}$ and $u_{a} (d + η_{a}) \to u_{a} (d) = v (d)$ as $a \to \infty$ . Furthermore on the basis of that work we quickly prove that (ii) for every choice of the above sequence $u_{a} (\cdot)$ , calling $Σ_{a}$ the system $Σ$ added with the frictionless constraint $u = u_{a} (t)$ and assuming $(\bar{q}, \bar{p})$ to be $Σ_{a}$ 's state at $t = d$ , along $Σ_{a}$ 's subsequent motion we have that $q (t) \in B (\bar{q}, 1 / a)$ $\forall t \in [d, d + η_{a}]$ and $\dot{q} (d + η_{a}) > a$ . Thus, for values of $a (\in N)$ large enough, $Σ_{a}$ 's motion has bursting characters.

In questa Nota (cui farà seguito una seconda) si considera un sistema Lagrangiano $Σ$ (eventualmente privo di Lagrangiano) riferito a $N + 1$ coordinate $q_{1} \dots, q_{N}$ , $u$ , con $u$ da usarsi come controllo e precisamente per aggiungere a $Σ$ un vincolo liscio del tipo $u = u (t)$ . I vincoli (lisci) di $Σ$ siano rappresentati nello spazio di Hertz dalla varietà $V_{t}$ (generalmente mobile). Si considera pure un istante $d$ (da usarsi per certe «proprietà di discontinuità al limite»), un punto $(\bar{q}, \bar{u})$ di $V_{d}$ , un valore $\bar{p}$ per il momento di $Σ$ coniugato a $q$ , e infine un controllo continuo $v (\cdot)$ con $v (d) = \bar{u}$ . Inoltre si suppone $\neq 0$ una certa quantità determinata dall'energia cinetica e dalle forze attive di $Σ$ , queste forze essendo supposte al più lineari in $\dot{u}$ . Un lavoro puramente matematico di Favretti ci permette di mostrare rapidamente che (i) $v (\cdot)$ è il limite in $C^{0}$ di una sequenza $u_{a} (\cdot)$ di controlli continui che hanno carattere di salto e salto $j_{a}$ $(= u_{a} (d + η_{a}) — \bar{u})$ in qualche intervallo $[d, d + η_{a}]$ e inoltre soddisfano certe condizioni, tra le quali che si abbia: $η_{a} \to 0^{+}$ , $j_{a} \to 0$ e $u_{a} (d + η_{a}) \to u_{a} (d) = v (d)$ . Inoltre sulla base di quel lavoro dimostriamo rapidamente che (ii) per ogni scelta della suddetta sequenza $u_{a} (\cdot)$ , detto $Σ_{a}$ il sistema $Σ$ soggetto all'addizionale vincolo liscio $u = u_{a} (t)$ e supposto che a $t = d$ $Σ_{a}$ sia nello stato $(\bar{q}, \bar{p})$ , lungo il susseguente moto di $Σ_{a}$ si ha che $q (t) \in B (\bar{q}, 1 / a)$ $\forall t \in [d, d + η_{a}]$ e $\dot{q} (d + η_{a}) > a$ . Così, per valori di $a (\in N)$ abbastanza alti, il moto di $Σ_{a}$ ha carattere di scoppio.

Publié le : 1991-12-01

MR 1152638 | Zbl 0784.70025

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     author = {Aldo Bressan and Marco Favretti},
     title = {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},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {2},
     year = {1991},
     pages = {339-343},
     zbl = {0784.70025},
     mrnumber = {1152638},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1991_9_2_4_339_0}
}

Bressan, Aldo; Favretti, Marco. 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. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 2 (1991) pp. 339-343. http://gdmltest.u-ga.fr/item/RLIN_1991_9_2_4_339_0/

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