Fractional infinite-horizon optimal control problems with a feed forward neural network scheme

Area of Science:

• Control Theory
• Applied Mathematics
• Computational Intelligence

Background:

• Fractional calculus extends classical calculus, offering more accurate models for complex systems.
• Optimal control problems (OCPs) are crucial for designing efficient system dynamics.
• Fractional infinite-horizon optimal control problems (FIHOCPs) present significant analytical and computational challenges.

Purpose of the Study:

• To develop a novel numerical method for solving FIHOCPs with Caputo fractional derivatives.
• To leverage neural networks for approximating solutions to these complex control problems.
• To demonstrate the efficacy of the proposed method through numerical examples.

Main Methods:

• Approximation of Caputo fractional derivatives with integer-order derivatives.
• Transformation of the infinite-horizon OCP into a finite-horizon OCP.
• Application of the Pontryagin minimum principle (PMP) to formulate an unconstrained minimization problem.
• Utilizing two-layered perceptron neural networks to construct trial solutions for state, costate, and control functions.

Main Results:

• The proposed neural network-based method effectively solves fractional infinite-horizon optimal control problems.
• The transformation and PMP formulation simplify the problem for neural network optimization.
• Numerical examples validate the accuracy and applicability of the developed technique.

Conclusions:

• The neural network approach provides a viable and efficient tool for addressing FIHOCPs.
• This method offers a practical way to handle complex fractional-order control systems.
• The study contributes to the advancement of computational methods in fractional optimal control.