Lie Theory and Control Systems Defined on Spheres

Abstract

We show in this paper that in constructing a theory for the most elementary class of control problems defined on spheres, some results from Lie theory play a natural role. In particular to understand controllability, optimal control, and certain properties of stochastic equations, Lie theoretic ideas are needed. The framework considered here is probably the most natural departure from the usual linear system/vector space problems which have dominated the control systems literature. For this reason our results are compared with those previously available for the finite-dimensional vector space case.