Approximate Solution of Variable-Order Fractional Optimal Control Problems

Abstract

This paper introduces a robust and highly accurate numerical scheme for solving a general class of variable-order fractional optimal control (VOFOC) problems based on the Caputo definition. The proposed method leverages a shifted Legendre spectral collocation (SLSC) technique, which effectively combines the superior approximation properties of global polynomials with the precision of Gaussian quadrature. The core of our approach lies in the novel derivation of an operational matrix for the variable-order fractional derivative (VOFD) of shifted Legendre polynomials. This matrix allows for the exact representation of the VOFDof the state variable, thereby transforming the complex fractional dynamical system constraint into a computationally tractable system of algebraic equations. The state and control variables are approximated by finite expansions of shifted Legendre polynomials with unknown coefficients. The objective functional is discretized using the shifted Legendre-Gauss-Lobatto (SLGL) quadrature rule, which is exact for polynomials of sufficiently high degree. Consequently, the original continuous-time VOFOC problem is converted into a nonlinear programming (NLP) problem, which can be efficiently solved using standard optimization solvers. A rigorous convergence analysis is provided, establishing that under mild conditions, the sequence of approximate solutions generated by our method uniformly converges to the exact optimal solution of the original VOFOC problem. The efficacy, versatility, and superiority of the proposed framework are demonstrated through six comprehensive numerical examples, including a practical application to train motion control. Results show that our method achieves exponential convergence rates, outperforms existing techniques like the variational iteration method, wavelet methods, and other spectral schemes in terms of accuracy, and yields solutions very close to the exact ones even with a small number of basis functions. The method proves to be a powerful tool for handling the complexities inherent in variable-order fractional calculus, offering a blend of theoretical robustness and computational efficiency.

Keywords: Variable-order fractional derivatives; Variable-order fractional optimal control problems; Spectral-collocation method;
Nonlinear programming.

2010 Mathematics Subject Classification. 26A25; 26A35