Optimal control of under-actuated systems with application to Lie groups

In this paper we study a class of optimal control problems known as the τ-elastic variational problem for second order, under-actuated systems. After introducing and stating the problem, we derive the necessary optimality conditions using two approaches. The first approach is purely variational where the resulting necessary conditions are represented by a single fourth order differential equation. In the second approach, we use the Lagrange multiplier technique. In this case, the necessary conditions are represented by a set of four first order differential equations. We show that the two results are equivalent. Finally, we further specialize the result for the compact semi-simple Lie group case and use SO(3) as an example. We also make some remarks on the SE(3) case, which is the subject of current research.