Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends

In this paper, we studied the necessary conditions for seeking optimal trajectory, optimal control, optimal time and optimal state. Then, we applied these qualitative properties to explore two numerical methods of a variable order fractional optimal control problem until full free ends that the initial time, initial state, terminal time and terminal state are free simultaneously. Based on the relations between variable order fractional calculus operators, i.e., the integral formulas by parts, the necessary conditions of optimality of a variable order fractional optimal control problem until full free ends drove from the Euler-Lagrange equation, applying Hamiltonian and variational principle. To develop the solution methods, we proposed the “optimization first, then discretization” (OFTD) method and the “discretization first, then optimization” (DFTO) method. The OFTD method is to solve the variable order fractional optimal control problem until full free ends by transforming the Euler-Lagrange equation into a nonlinear system using the Grünwald-Letnikov definition and manipulating the transversal conditions as an objective function of error quadratic minimization, i.e., a nonlinear programming type problem with equality constraints. The DFTO method is to solve the problem by transforming the variable order fractional calculus into a classical optimal control problem with integer order using the expansion formulas of the variable order fractional calculus operators. Finally, we demonstrated the validity and accuracy of the proposed methods through various types of numerical test problems.