Some Results on Stabilizability of Controlled Lagrangian Systems by Energy Shaping

We provide necessary and sufficient conditions for Lyapunov stabilizability and exponential stabilizability by the energy shaping method for the class of all linear controlled Lagrangian systems of an arbitrary degree of under-actuation, and for the class of all controlled Lagrangian systems of one degree of under-actuation. We give a sufficient condition for asymptotic stabilizability for the class of all controlled Lagrangian systems of one degree of under-actuation. For a general controlled Lagrangian system, we give only necessary conditions for Lyapunov stabilizability and exponential stabilizability by energy shaping. In addition, we make a new derivation of the Euler-Lagrange matching conditions both in a simple tensor form and in a coordinate-dependent form, for which we make effective use of gyroscopic forces.