Single-input control systems on the Euclidean group SE(2)

Abstract

We consider a general single-input left-invariant control affine system, evolving on the Euclidean group SE(2). Any such controllable control system is (detached feedback) equivalent to one of two typical cases. In each case, we consider an optimal control problem (with quadratic cost) which is then lifted, via the Pontryagin Maximum Principle, to a Hamiltonian system on the dual space se(2)∗. The reduced Hamilton equations are derived and the stability nature of all equilibrium states is then investigated. Finally, these equations are explicitly integrated by elliptic functions.