Necessary optimality conditions for two dimensional fractional optimal control problems and error estimates for the numerical approximation

This paper presents the necessary optimality conditions and a new method for solving a class of two dimension fractional optimal control problem (FOCP-2D) based on shifted Legendre polynomials. By using the Lagrange multiplier method and integration by part formula within the calculus of variations, the necessary optimality conditions are derived as two-point fractional-order boundary value problem. Fractional operators of shifted Legendre polynomial are computed and the necessary optimality conditions have been converted into a system of algebraic equations by utilizing these fractional operators. L-2-error estimates in the approximation of a function and its fractional derivative by shifted Legendre polynomial have been derived. Moreover, convergence analysis of the proposed method has also been discussed and in the last, illustrative examples are included to demonstrate the effectiveness and applicability of the proposed method.