Generalized synthesis and computation of nonclosed accessible sets in control theory

Accessible sets are important invariants of control systems, and computation of boundary points of accessible sets plays important roles in the solution of optimal control problems, and other classical problems like motion planning or trajectory tracking. Pontryagin's Maximum Principle is an important tool to compute boundary points of accessible sets. However, it's usefulness is severely reduced when the accessible set is not closed. We discuss an approach to overcome these difficulties in the class of control-affine systems and give a geometric characterization of so-called "generalized extremals". The computations required to obtain generalized extremals by this method are considerably simplified with respect to alternative approaches.