Warga's abundant sets and their role in optimal control problems

Warga's concept of an abundant set and the related controllability-extremality alternative theorem provides a powerful framework for deriving necessary optimality conditions for various optimal control problems. This method enables us to focus on the essential issue of proving the set of original controls is abundant in an appropriately constructed relaxed control set. This abundant property is usually a strengthened version of the Filippov-Wazewski relaxation theorem. It has been applied to many optimal control problems and is a method with great potential.