Solvability and Optimal Control Problem for Coupled Fractional Differential Equations with Impulsive Effects

In this paper, a coupled system of nonlinear impulsive fractional differential equations with variable coefficient nonlinear term is considered. At first, we investigate the solvability of coupled system. By using a fixed point theorem of generalized concave increasing operator, the existence and uniqueness result of positive solutions for each given control functions are obtained. Secondly, we focus on the optimal control problem of the positive solutions to the system, and then, we prove the existence of an optimal control and its stability that minimizes the nonlinear cost functional. Finally, a related example is given to illustrate the effectiveness of the abstract results.