A new hybrid method for two dimensional nonlinear variable order fractional optimal control problems

This paper presents an efficient operational matrix method for two dimensional nonlinear variable order fractional optimal control problems (2D-NVOFOCP). These problems include the nonlinear variable order fractional dynamical systems (NVOFDS) described by partial differential equations such as the diffusion-wave, convection-diffusion-wave and Klein-Gordon equations. The variable order fractional derivative is defined in the Caputo type. The proposed hybrid method is based on the transcendental Bernstein series (TBS) and the generalized shifted Chebyshev polynomials (GSCP). The new operational matrices of derivatives are generated for the mentioned polynomials. The state and control functions are expressed by the TBS and GSCP with free coefficients and control parameters. These expansions are substituted in the performance index and the resulting operational matrices are employed to extract algebraic equations from the approximated NVOFDS. The constrained extremum is obtained by coupling the algebraic constraints from the dynamical system and the initial and boundary conditions with the algebraic equation extracted from the performance index by means of a set of unknown Lagrange multipliers. The convergence analysis is discussed and several numerical experiments illustrate the efficiency and accuracy of the proposed method.