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

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 2 (1991) no. 4, pp. 339-343.

Abstract (VO)

In this Note (which will be followed by a second) we consider a Lagrangian system \( \Sigma \) (possibly without any Lagrangian function) referred to \( N + 1 \) coordinates \( q_{1} \cdots , q_{N} \), \( u \), with \( u \) to be used as a control, and precisely to add to \( \Sigma \) a frictionless constraint of the type \( u = u(t)\). Let \( \Sigma \)'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 \( \Sigma \)'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 \( \Sigma \)'s kinetic energy and \( \Sigma \)'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 \(\left[ d, d+ \eta_{a} \right] \) and satisfy certain conditions including that both \( \eta_{a} \to 0^{+} \) and \( u_{a} (d+\eta_{a}) \to u_{a}(d)= v(d) \) 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 \( \Sigma_{a} \) the system \( \Sigma \) added with the frictionless constraint \( u = u_{a}(t) \) and assuming \( (\bar{q},\bar{p}) \) to be \( \Sigma_{a} \)'s state at \( t = d \), along \( \Sigma_{a} \)'s subsequent motion we have that \( q(t) \in B(\bar{q}, 1/a) \)\( \forall t \in \left[ d,d+\eta_{a} \right] \) and \( \dot{q}(d +\eta_{a})>a \). Thus, for values of \( a(\in \mathbb{N}) \) large enough, \( \Sigma_{a} \)'s motion has bursting characters.

Riassunto

In questa Nota (cui farà seguito una seconda) si considera un sistema Lagrangiano \( \Sigma \) (eventualmente privo di Lagrangiano) riferito a \( N + 1 \) coordinate \( q_{1} \cdots , q_{N} \), \( u \), con \( u \) da usarsi come controllo e precisamente per aggiungere a \( \Sigma \) un vincolo liscio del tipo \( u = u(t)\). I vincoli (lisci) di \( \Sigma \) 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 \( \Sigma \) 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 \( \Sigma \), 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 \( j_{a} \)\( (= u_{a}(d+ \eta_{a}) — \bar{u}) \) in qualche intervallo \( [d,d+\eta_{a}] \) e inoltre soddisfano certe condizioni, tra le quali che si abbia: \( \eta_{a} \to 0^{+} \), \( j_{a} \to 0 \) e \( u_{a} (d+\eta_{a}) \to u_{a}(d)= v(d) \). Inoltre sulla base di quel lavoro dimostriamo rapidamente che (ii) per ogni scelta della suddetta sequenza \( u_{a} (\cdot) \), detto \( \Sigma_{a} \) il sistema \( \Sigma \) soggetto all'addizionale vincolo liscio \( u = u_{a}(t) \) e supposto che a \( t = d \)\( \Sigma_{a} \) sia nello stato \( (\bar{q},\bar{p}) \), lungo il susseguente moto di \( \Sigma_{a} \) si ha che \( q(t) \in B(\bar{q}, 1/a) \)\( \forall t \in \left[ d,d+\eta_{a} \right] \) e \( \dot{q}(d +\eta_{a})>a \). Così, per valori di \( a(\in \mathbb{N}) \) abbastanza alti, il moto di \( \Sigma_{a} \) ha carattere di scoppio.

Bibliographie

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