Some second-order necessary conditions in optimal control

In optimal control problems any extremal arc which trivially satisfies the Maximum Principle, that is a first-order control variation produces no change in cost, is called singular. Higher-order conditions are then needed to check the optimality of such arcs. Using the Volterra series associated with the variation of the cost functional gives a new context for analyzing singular optimal control problems. A basic optimality criterion for a fixed terminal time Mayer problem is obtained which allows one to derive the necessary conditions for optimality in terms of Lie brackets of vector fields associated with the dynamics of the problem.