Singular solutions in optimal control: second order conditions and a shooting algorithm

In this article we study optimal control problems for systems that are affine in one part of the control variable. Finitely many equality and inequality constraints on the initial and final values of the state are considered. We investigate singular optimal solutions for this class of problems. First, we obtain second order necessary and sufficient conditions for weak optimality. Afterwards, we propose a shooting algorithm and show that the sufficient condition above-mentioned is also sufficient for the local quadratic convergence of the algorithm.