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On the application of control theory to certain problems for Lagrangian systems, and hyperimpulsive motion for these. I: Some general mathematical considerations on controllable parameters. (English) Zbl 0669.70029
In applying control (or feedback) theory to (mechanics) 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 $${\mathcal O}_{{\mathcal R}}$$ 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 controllable, in the sense that one can satisfactorily treat extremum problems concerning a class $$\Gamma_{{\tilde \gamma},\Delta,\Delta '}$$ of controls $$\gamma ={\hat \gamma}(t)={\tilde \gamma}[u(t)]$$ 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($$\cdot)\in {\mathcal L}^ 1(\Delta,\Delta ')$$ where $$\Delta$$ and $$\Delta '$$ are suitable compact segments of $$\mathbb{R}$$ ($$\overset\circ\Delta \neq \emptyset \neq \overset\circ\Delta')$$, and (iv) are physically carried out in the above way.
One of the aims of another author’s work [(*) Hyper-impulsive motions and controllable co-ordinates for Lagrangian systems (to appear)] is just to write the above conditions (C) by using some recent results in control theory - see the author, (**) Rend. Semin. Mat. Univ. Padova 78, 227 ff. (1987), where H. J. Sussmann’s paper [Ann. Probab. 6, 19-41 (1978; Zbl 0391.60056)] is extended from continuous to measurable controls - and some consequences of them presented by the author in: (***) “On some recent results in control theory, for their application to Lagrangian systems” (to appear). The present work, divided into the Notes I to III, has in part the role of an abstract, in that some of the author’s cited works have not yet been proposed for publication and e.g. conditions (C) are written in Note II, without proof. In Note I conditions (C) are shown to be necessary for the last M co-ordinates of $$\chi$$ to be 1- dimensionally controllable; in doing this proof, this controllability 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 should 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 (**), by (also) using some results obtained in (***). The work (*) also aims at extending the well known theory of impulsive motions, with continuous positions but with velocities suffering first oder discontinuities, to a theory of hyper-impulsive motions, in which positions also suffer such discontinuities. In case the components $${\mathcal O}_{{\mathcal R}}$$ depend on Lagrangian velocities in a certain way, in Note II - see its Summary - one proves some analogues for jumps of the results on controllability stated in Note I.

##### MSC:
 70Q05 Control of mechanical systems 70H03 Lagrange’s equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 93C20 Control/observation systems governed by partial differential equations 93C95 Application models in control theory