Legendre spectral-collocation method for solving some types of fractional optimal control problems.

Area of Science:

• * Numerical Analysis
• * Optimal Control Theory
• * Fractional Calculus

Background:

• * Fractional Optimal Control Problems (FOCPs) present unique challenges in finding efficient solutions.
• * The Caputo definition of fractional derivatives is commonly used in modeling dynamic systems.
• * Spectral-collocation methods offer a powerful framework for approximating solutions to differential equations.

Purpose of the Study:

• * To develop and present approximate solution techniques for fractional optimal control problems.
• * To apply the Legendre spectral-collocation method to FOCPs with Caputo fractional derivatives.
• * To demonstrate the validity and applicability of the proposed numerical methods.

Main Methods:

• * Legendre spectral-collocation method applied to FOCPs.
• * Approximation of necessary optimality conditions using the Hamiltonian.
• * Discretization of state equations via the trapezoidal rule and Rayleigh-Ritz method.

Main Results:

• * Two distinct approaches using the Legendre spectral-collocation method were successfully implemented.
• * The proposed methods yielded approximate solutions for the FOCPs.
• * Illustrative examples confirmed the effectiveness and applicability of the techniques.

Conclusions:

• * The Legendre spectral-collocation method provides a viable approach for solving FOCPs.
• * The presented techniques are effective for approximating solutions in fractional optimal control.
• * The study validates the use of these numerical methods for FOCPs with Caputo derivatives.