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.

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The objective of our paper is to investigate the optimal control of semilinear population dynamics system with diffusion using semigroup theory. The semilinear population dynamical model with the nonlocal birth process is transformed into a standard abstract semilinear control system by identifying the state, control, and the corresponding function spaces. The state and control spaces are assumed to be Hilbert spaces. The semigroup theory is developed from the properties of the population operators and Laplacian operators. Then the optimal control results of the system are obtained using the C0-semigroup approach, fixed point theorem, and some other simple conditions on the nonlinear term as well as on operators involved in the model.

This paper deals with fractional optimal control governed by semilinear equations using the increment approach. We have considered controlled object as with the initial conditions: where α ∈ (1, 2], s(τ) is state variable in , B(τ) ∈ Lp, ϱ(0, T), and . Let the m‐dimensional control vector function be C(τ) and defined as . Assume the function η(τ, C(τ)) satisfies Caratheodory condition and defined on . We have obtained our results with the help of the adjoint equation and Pontryagin's maximum condition. For better understanding, we have included one example.