In this paper, we apply the classical control theory to a fractional differential system in a bounded domain. The fractional optimal control problem (FOCP) for differential system with time delay is considered. The fractional time derivative is considered in a Riemann-Liouville sense. We first study the existence and the uniqueness of the solution of the fractional differential system with time delay in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a FOCP is considered as a function of both state and control variables, and the dynamic constraints are expressed by a partial fractional differential equation. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of a right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in detail.MSC: 46C05; 49J20; 93C20

In this paper, the fractional optimal control problem for differential system is considered. The fractional time derivative is considered in Riemann-Liouville sense. Constraints on controls are imposed. Necessary and sufficient optimality conditions for the fractional Dirichlet and Neumann problems with the quadratic performance functional are derived. Some examples are analyzed in details.

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In this paper, we apply the classical control theory to a variable order fractional differential system in a bounded domain. The Fractional Optimal Control Problem (FOCP) for variable order differential system is considered. The fractional time derivative is considered in a Caputo sense. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the variable order fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE) with variable order. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.

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In this paper, by using the Dubovitskii-Milyutin theorem, we consider a differential inclusions problem with fractional-time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. The Atangana-Baleanu fractional derivative of order α in the sense of Caputo with respect to time t, is considered. Existence and uniqueness of solution are proved by means of the Lions-Stampacchia theorem. The existence of solution is obtained for all values of the fractional parameter α ∈ (0, 1). Moreover, by applying control theory to the fractional differential inclusions problem, we obtain an optimality system which has also a unique solution. The controllability of the fractional Dirichlet problem is studied. Some examples are analyzed in detail. MSC: 46C05; 49J20; 93C20