Optimal Control on Nilpotent Lie Groups

Let G be a nilpotent Lie group and let δ = {X1,X2} be a bracket generating left invariant distribution on G. In this paper we study the left invariant optimal control problem on G defined by the differential equation ġ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left( t \right) = u_1 \left( t \right)X_1 \left( {g\left( t \right)} \right) + u_2 \left( t \right)X_2 \left( {g\left( t \right)} \right),$$ \end{document} the cost functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Lambda \left( {g,u} \right) = \frac{1}{2}\int {\left( {u_1^2 + u_2^2 } \right)dt} ,$$ \end{document} and the family of measurable and bounded control functions t → u = (u1(t), u2(t)). We use the Pontryagin maximum principle and the associated Hamiltonian formalism to obtain the optimal controls of the system. Optimal solutions are necessarily projections of trajectories of a Hamiltonian system which in the normal case can be explicitly integrated in terms of hyperelliptic functions. Abnormal extremals (those which do not depend on the cost functional) turn out to be not strictly abnormal.