Fractional Optimal Control For Infinite Variables Parabolic SystemsWith Time Lags Given In Integral Form

In this paper, the fractional optimal control problem for distributed parabolic systems involving constant lags in the integral form both in the state equations and in the boundary condition is considered. The fractional time derivative is considered in a Caputo sense. The system contains model operator of second order with infinite number of variables. 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 (PFDE). 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 fractional optimal control. Some specific properties of the optimal control are discussed. Some examples are analyzed in details.