An Operational Matrix Method Based on the Gegenbauer Polynomials for Solving a Class of Fractional Optimal Control Problems

One of the most important classes of fractional calculus is the fractional optimal control problem (FOCP), which arises in engineering. This study presents a direct and efficient numerical method for solving a class of (FOCPs) in which the fractional derivative is in the Caputo sense and the dynamic system includes the fractional- and integer-order derivatives. For this purpose, we use the operational matrix of fractional Riemann-Liouville integration based on the shifted Gegenbauer polynomials. First, the fractional- and integer-order derivatives in the given problem are approximated based on the shifted Gegenbauer polynomials with unknown coefficients. Then by substituting these approximations and the equation derived from the dynamic constraint into the cost functional, an unconstrained optimization problem is obtained. The main advantage of this approach is that it reduces the FOCP given to an unconstrained optimization problem and using the necessary optimality conditions yields a system of algebraic equations which can be easily solved by Newton’s iterative method. In addition, the convergence of the method is proved via several theorems. Finally, some numerical examples are presented to illustrate the validity and applicability of the proposed technique.

Keywords:

Caputo fractional derivative , Numerical method , Optimal control problems , Riemann-Liouville fractional integration , Shifted Gegenbauer polynomials operational matrix