Optimality conditions and duality for a non-linear time-delay control problem

Various types of sufficient conditions of optimality for non-linear optimal control problems with delays in state and control variables are formulated. The involved functions are not required to be convex. A second-order sufficient condition is shown to be related to the existence of solutions of a Riccati-type matrix differential inequality. Under certain concavity and convexity assumptions, dual problems are constructed without using convex conjugates.