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In this article, we study a homogeneous infinite order Dirichlet and Neumann boundary fractional equations in a bounded domain. The fractional time derivative is considered in a Riemann-Liouville sense. Constraints on controls are imposed. The existence results for equations are obtained by applying the classical Lax-Milgram Theorem. The performance functional is in quadratic form. Then we show that the optimal control problem associated to the controlled fractional equation has a unique solution. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of the right fractional Caputo derivative, we obtain an optimality system. The obtained results are well illustrated by examples.

Fractional optimal control problem for infinite order system with control constraints