A relaxation approach to optimal control of Volterra integral equations

We consider an optimal control problem for systems defined by nonlinear Volterra integral equations, with state constraints. No convexity assumptions are made on the data, and the problem is transformed into its relaxed form. We prove the existence of an optimal relaxed control and derive necessary conditions for optimality in the form of a relaxed minimum principle of Pontryagin type. We then apply a mixed Frank-Wolfe penalty method which constructs sequences of relaxed controls converging to extremal controls for this problem. A numerical example is given.