Pontryagin principle for a Mayer problem governed by a delay functional differential equation

We establish Pontryagin principles for a Mayer’s optimal control problem governed by a functional differential equation. The control functions are piecewise continuous and the state functions are piecewise continuously differentiable. To do that, we follow the method created by Philippe Michel for systems governed by ordinary differential equations, and we use properties of the resolvent of a linear functional differential equation.