Solving time delay fractional optimal control problems via a Gudermannian neural network and convergence results

Area of Science:

• Control Theory
• Applied Mathematics
• Computational Intelligence

Background:

• Optimal control problems (OCPs) are crucial in engineering and science.
• Fractional-order systems and time delays introduce significant complexity to OCPs.
• Existing methods may struggle with the analytical and numerical challenges posed by these systems.

Purpose of the Study:

• To develop a novel Gudermannian neural network (GNN) scheme for solving OCPs of fractional-order systems with state and control delays.
• To address the challenges associated with fractional derivatives (Caputo sense) and time-delayed arguments.
• To provide an effective numerical approach for approximating solutions to the Hamiltonian conditions derived from the Pontryagin minimum principle.

Main Methods:

• Transformation of the time-delayed problem into an equivalent one without delays using Padé approximation.
• Approximation of states, Lagrange multipliers, and control functions using trial solutions constructed with two-layered perceptrons.
• Minimization of an error function via an unconstrained optimization scheme to determine unknown weights and biases of the neural network.

Main Results:

• The proposed Gudermannian neural network scheme successfully approximates solutions for fractional-order optimal control problems with delays.
• Numerical examples demonstrate the effectiveness and accuracy of the GNN method in solving these complex control problems.
• The method provides a viable alternative for tackling challenging optimal control scenarios in fractional dynamics.

Conclusions:

• The Gudermannian neural network scheme offers a powerful and efficient tool for solving optimal control problems in fractional-order systems with time delays.
• The integration of Padé approximation and neural network techniques provides a robust framework for handling system complexities.
• This research contributes to the advancement of computational methods for fractional optimal control.