A fractional optimal control problem with final observation governed by wave equation

Area of Science:

• Applied Mathematics
• Fractional Calculus
• Optimal Control Theory

Background:

• Fractional wave equations model various physical phenomena.
• Optimal control problems require determining control functions to minimize a cost functional.
• Existing methods for fractional wave equations lack efficient numerical approximations.

Purpose of the Study:

• To investigate the optimal control problem for a fractional wave equation.
• To establish the existence and uniqueness of an optimal solution.
• To develop an efficient numerical approximation for the fractional wave equation.

Main Methods:

• Formulation of the fractional optimal control problem.
• Calculation of the Frechet derivative of the cost functional using an adjoint problem.
• Derivation of necessary optimality conditions.
• Introduction of a numerical approximation for the fractional wave equation with Atangana-Baleanu derivative.

Main Results:

• The existence and uniqueness of an optimal solution for the fractional optimal control problem are proven.
• Necessary optimality conditions are derived.
• An efficient numerical approximation for the fractional wave equation is presented.

Conclusions:

• The study successfully formulates and solves an optimal control problem for fractional wave equations.
• The derived optimality conditions and numerical methods provide a foundation for further research in this area.
• The Atangana-Baleanu derivative offers a promising approach for modeling and simulation.