Control and differential geometry for mechanical systems


Lunedì 24 novembre alle ore 11 in aula 2AB/45 il professor Franco Rampazzo terrà un seminario dal titolo "Control and differential geometry for mechanical systems".

Il seminario sarà preceduto da un prologo di introduzione all'argomento del seminario che si terrà dalle 10 alle 10.30 in Sala Riunioni VII piano, seguito da un momento di pausa e conversazione con caffè e biscotti in Common Room.

Let us consider a (N +M)-dimensional mechanical system by state coordinates (q1;.....;q^N; u1; ...; u^M), and let us regard some of the latter, say (u1;..... ; u^M), as controls. This reduces the dimension of the system to N. For instance, two rigidly connected material points {P1; P2} in the plane give rise to a three-dimensional system. Yet, if we prescribe the motion of P1, {P1; P2}
becomes a one-dimensional system. Let us remark that the predetermination of the u^j's evolutions is usually referred to as a moving (holonomic) constraint. In the general case, the dynamics governing the motions of the q^i's - besides containing the parameters u^j- depends quadratically on the derivatives of u^j as well. Therefore, it is natural to pursue a careful investigation of the solutions' set for an ODE quadratically depending on (unbounded) controls. At the same time, some remarkable differential geometric aspects have to be considered. In particular, the latter involve the kinetic metric and a curvature term related to the leaves {u = cost}. As a simple instance, the non-vanishing of this curvature explains the well-known stability of the inverted pendulum whose pivot is subject to rapid vertical oscillations.

Rif. int. M. Bardi, F. Cardin, M. Putti, F. Rampazzo