Control Foliations for Mechanical Systems

The idea of regarding the last M parameters u = (u1, … , uM) of a system of Lagrangian coordinates (q1, … , qN , u1, … , qM) as controls is made intrinsic by considering a foliation of the configuration space Q whose leaves are locally represented by the equations u = constant. A control u(·) is then a path in the space of leaves. We review some results concerning the way the control equations governing the motion on the leaves depend on the derivative We allow system to be subject to a non holonomic constraint ( , ) ∈ γ as well, where γ is a (non-integrable) distribution of codimension ν. In general, these equations, whose state variables are represented by (q1, … , qN) and suitable momenta (ξ1, … , ξN–ν), are quadratic polynomials of (with coefficients depending on q, ξ, and u). The quadratic term turns out to be interesting for controllability and (vibrational) stabilizability purposes. On the other hand the special case when the quadratic term is vanishing characterizes the possibility to well pose the equations with not regular -even discontinuous- controls u.