Linear quadratic regulator problem governed by granular neutrosophic fractional differential equations

Quadratic cost functions estimation in the linear optimal control systems governed by differential equations (DEs) or partial differential equations (PDEs) has a well-established discipline in mathematics with many interfaces to science and engineering. During its development, the impact of uncertain phenomena to objective function and the complexity of the systems to be controlled have also increased significantly. Many engineering problems like magnetohydromechanical, electromagnetical and signal analysis for the transmission and propagation of electrical signals under uncertain environment can be dealt with. In this paper, we study the optimal control problem with operating a fractional DEs and PDEs at minimum quadratic objective function in the framework of neutrosophic environment and granular computing. However, there has been no studies appeared on the neutrosophic calculus of fractional order. Hence, we will introduce some derivatives of fractional order, including the neutrosophic Riemann-Liouville fractional derivatives and neutrosophic Caputo fractional derivatives. Next, we propose a new setting of two important problems in engineering. In the first problem, we investigate the numerical and exact solutions of some neutrosophic fractional DEs and neutrosophic telegraph PDEs. In the second problem, we study the optimality conditions together with the simulation of states of a linear quadratic optimal control problem governed by neutrosophic fractional DEs and PDEs. Some key applications to DC motor model and one-link robot manipulator model are investigated to prove the effectiveness and correctness of the proposed method.