A note on the existence and optimal control for mixed Volterra–Fredholm-type integrodifferential dispersion system of third order

The optimal control problem consists of a performance index subject to a set of differential equations that describes the path of the control and state variables. The main aim of this article is to prove the existence and uniqueness of a mild solution, optimal control, and time-optimal control of a mixed Volterra–Fredholm-type third-order dispersion system. By applying the strongly continuous semigroup theory and the Banach fixed-point theorem, we prove the existence and uniqueness of the considered system. The optimal control results are proved by using Mazur's lemma, Gronwall's inequality, and the minimizing sequence technique. The discussion on the time-optimal control of the third-order dispersion system is also presented.