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

Résumé

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.