The inverse problem of the calculus of variations and the stabilization   of controlled Lagrangian systems
  We apply methods of the so-called `inverse problem of the calculus of variations' to the stabilization of an equilibrium of a class of two-dimensional controlled mechanical systems. The class is general enough to include, among others, the inverted pendulum on a cart and the inertia wheel pendulum. By making use of a condition that follows from Douglas' classification, we derive feedback controls for which the control system is variational. We then use the energy of a suitable controlled Lagrangian to provide a stability criterion for the equilibrium. 
M. Farr\'e Puiggal\'i and T. Mestdag
math-ph math.DS math.MP math.OC