On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters

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. 91-105

Résumé

In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control $u(\cdot)$. However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control $u=u(\cdot)$, where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system $\Sigma$, referred to a system $\chi$ of Lagrangian co-ordinates, and to try and write some handy conditions, (C), on the coefficients of $\Sigma$'s kinetic energy $\mathcal{T}$ and the Lagrangian components $\mathcal{Q}_{\mathcal{R}}$ of the forces applied to $\Sigma$, at least sufficient to satisfactorily use controls of the second kind. More specifically the conditions (C) sought for, should imply that the last M coordinates in $\chi$ are 1-dimensionally controllizable, in the sense that one can satisfactorily treat extremum problems concerning a class $\Gamma_{\tilde{\gamma},\Delta,\Delta^{\prime}}$, of controls $\gamma = \hat{\gamma}(t) = \tilde{\gamma} \left[u(t)\right]$ that (i) take as values $M$-tuples of values of those co-ordinates, (ii) have the same arbitrarily prefixed $C^{2}$-path $\tilde{\gamma}$ as trajectory, (iii) are Lebesgue integrable in that $u(\cdot) \in \mathfrak{L}^{1} (\Delta,\Delta^{\prime})$ where $\Delta$ and $\Delta^{\prime}$ are suitable compact segments of $\mathbb{R}(\mathring{\Delta} \ne \emptyset \ne \mathring{\Delta}^{\prime})$, and (iv) are physically carried out in the above way. One of the aims of [4] is just to write the above conditions (C) by using some recent results in control theory-see [2] where Sussmann's paper [7] is extended from continuous to measurable controls-and some consequences of them presented in [3]. The present work, divided into the Notes I to III, has in part the role of an abstract, in that the works [4] to [5] have not yet been proposed for publication and e.g. conditions (C) are written in Note II, i.e. [6], without proof. In Note I conditions (C) are shown to be necessary for the last $M$ coordinates of $\chi$ to be 1-dimensionally controllizable; in doing this proof, this controllizability is regarded to include certain (relatively weak) continuity properties, which are important for checking experimentally the theory being considered, and which (therefore) are analogues of the requirement that the solutions of (physical) differential systems shoud depend on the initial data continuously. Conversely conditions (C) imply that even stronger continuity properties hold for $(\Sigma,\chi,M)$. The above proofs are performed in Note I from the general (purely mathematical) point of view considered in [2], by (also) using some results obtained in ([2] and) [3]. The work [4] also aims at extending the well known theory of impulsive motions, with continuous positions but with velocities suffering first order discontinuities, to a theory of hyper-impulsive motions, in which positions also suffer such discontinuities. In case the components $\mathcal{Q}_{\mathcal{R}}$ depend on Lagrangian velocities in a certain way, in Note II-see its Summary-one proves some analogues for jumps of the results on controllizability stated in Note I. In Part III some intuitive justifications given in Part II are replaced with theorems; furthermore an invariance property is proved.