On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.

Bressan, Aldo

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 3, pp. 461-471

Résumé

In [1] I and II various equivalence theorems are proved; e.g. an ODE $(\mathcal{E}) \dot{z} = F(t,z,u,\dot{u}) \, (\in \mathbb{R}^{m})$ with a scalar control $u = u(\cdot)$ is linear w.r.t. $\dot{u}$ iff $(\alpha)$ its solution $z(u,\cdot)$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta)$$z(u, \cdot)$ satisfies certain conditions by which $1^{st}$-order discontinuities of $u$ and $\dot{u}$ can be treated satisfactorily. In the case when, for $z = (q, p)$ equation $(\mathcal{E})$ is a semi-Hamiltonian system, equivalent to a system of Lagrangian equations of a general type, the importance or compulsory character in many situations, of the conditions hinted at in $(\alpha)$ and $(\beta)$, have received some intuitive justifications in [1] II. In the present paper some of these are replaced by theorems and thus the importance of the above linearity is strengthened. E.g. this linearity is shown, roughly speaking, to follow from the continuity (in the afore-mentioned sense) of the function $u \vdash q ( u , \cdot)$ alone. In the above semi-Hamiltonian case, the linearity of equation $(\mathcal{E})$ w.r.t. $u$ is also proved to be invariant under certain transformations of Lagrangian co-ordinates.