A study of a class of invariant optimal control problems on the Euclidean group SE(2)

Abstract

The aim of this thesis is to study a class of left-invariant optimal control problems on the matrix Lie group SE(2). We classify, under detached feedback equivalence, all controllable (left-invariant) control affine systems on SE(2). This result produces six types of control affine systems on SE(2). Hence, we study six associated left-invariant optimal control problems on SE(2). A left-invariant optimal control problem consists of minimizing a cost functional over the trajectory-control pairs of a left-invariant control system subject to appropriate boundary conditions. Each control problem is lifted from SE(2) to T*SE(2) ≅ SE(2) x se (2)*and then reduced to a problem on se (2)*. The maximum principle is used to obtain the optimal control and Hamiltonian corresponding to the normal extremals. Then we derive the (reduced) extremal equations on se (2)*. These equations are explicitly integrated by trigonometric and Jacobi elliptic functions. Finally, we fully classify, under Lyapunov stability, the equilibrium states of the normal extremal equations for each of the six types under consideration.