Optimal Control Problems

In this thesis, we consider smooth optimal control systems that evolve on Lie groups. Pontryagin’s maximum principle allows us to search for local solutions of the optimal control problem by studying an associated Hamiltonian dynamical system. When the associated Hamiltonian function possess symmetries, we can often study the Hamiltonian system in a vector space whose dimension is lower than the original system. We apply these symmetry reduction techniques to optimal control problems on Lie groups for which the associated Hamiltonian function is left-invariant under the action of a subgroup of the Lie group. Necessary conditions for optimality are derived by applying Lie-Poisson reduction for semidirect products, a previously developed method of symmetry group reduction in the field of geometric mechanics. Our main contribution is a reduced sufficient condition for optimality that relies on the nonexistence of conjugate points. Coordinate formulae are derived for computing conjugate points in the reduced Hamiltonian system, and we relate these conjugate points to local optimality in the original optimal control problem. These optimality conditions are then applied to an example optimal control problem on the Lie group SE(3) that exhibits symmetries with respect to SE(2), a subgroup of SE(3). This optimal control problem can be used to model either a kinematic airplane, i.e. a rigid body moving at a constant speed whose angular velocities can be controlled, or a Kirchhoff elastic rod in a gravitational field.