Geometric Control Theory

Abstract

We survey the basic theory, results, and applications of geometric control theory. A control system is a dynamical system with parameters called controls or inputs. A control trajectory is a trajectory of the control system for a particular choice of the inputs. A control system is called controllable if every two points of the underlying space can be connected by a control trajectory. Two fundamental problems of control theory include: 1) Is the control system controllable? 2) If it is controllable, how can we construct an input to obtain a particular control trajectory? We shall investigate the first problem exclusively for affine drift free systems. A control system is affine if it is of the form: ẋ=X0(x)+u1X1(x)+...+ukXk(x) where X0 is the drift vector field, X1(x),...,Xk(x) are the control vector fields, and u1, ... , uk are the inputs. An affine system is called drift-free if X0=0. The fundamental theorem of control theory (known as Chow-Rashevsky theorem) states that an affine drift-free control system is controllable if the control vector fields together with their iterated Lie brackets span the entire tangent bundle of the underlying space. We prove this result in the simplest case when the space is 3-dimensional and k=2.