Mathematical problems in the control of underactuated systems (Q2752223)

This paper is about stabilization by feedback of a certain type of nonlinear underactuated systems (i.e., systems with fewer control inputs than degrees of freedom). The class of systems considered can be used to describe a wide range of physical systems. NEWLINENEWLINENEWLINEThe authors give brief accounts of both the linearization and the Lyapunov function approaches to stabilization, followed by a discussion of the ``matching'' method to obtain simultaneously a closed-loop control and a Lyapunov function for a given control system of the type under consideration. They review their method to solve the matching equations and give a special account of the results obtained in the two-dimensional case. NEWLINENEWLINENEWLINEIn a final section the authors discuss open problems related to the choice of ``good'' stabilizing control laws. The issues involved are the choice and computation of criteria to evaluate the ``size'' of the basin of attraction, the time necessary to approach the equilibrium and the ``cost'' of control laws.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00038].