Existence and optimal control problem for semilinear fractional order (1,2] control system

In this article, we have derived some sufficient conditions for existence of optimal control for semilinear control system of fractional order (1,2] in Hilbert space. We have discussed existence and uniqueness of mild solution with the help of Banach fixed point theorem for proposed problem. Under certain conditions stated, Lagrange's problem admits at least one optimal control pair. Finally, one example is given to understand theoretical results in better manner.