[Paper Review] Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel

Jean‐Daniel Djida, Gisèle Mophou+1 more|arXiv (Cornell University)|Nov 24, 2017

Fractional Differential Equations Solutions13 references3 citations

TL;DR

This paper establishes the existence and uniqueness of solutions for a fractional diffusion equation with a nonlocal, nonsingular Mittag-Leffler kernel fractional derivative in time, using spectral methods for all α ∈ (0,1). It further formulates and solves an optimal control problem via the Lagrange multiplier method, deriving a first-order optimality system that yields a unique control minimizing a quadratic cost functional.

ABSTRACT

In this paper, we consider a diffusion equation with fractional-time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. Existence and uniqueness of solution are proved by means of a spectral argument. The existence of solution is obtained for all values of the fractional parameter $α\in (0,1)$. Moreover, by applying control theory to the fractional diffusion problem we obtain an optimality system which has also a unique solution.

Motivation & Objective

• To establish existence and uniqueness of solutions for a fractional diffusion equation with Atangana-Baleanu time derivative for all α ∈ (0,1), overcoming the restriction α ∈ (1/2,1) seen in Caputo and Riemann-Liouville formulations.
• To formulate and analyze an optimal control problem for the fractional diffusion equation with the goal of minimizing a quadratic cost functional involving state tracking and control effort.
• To derive the first-order optimality system using the Lagrange multiplier method, ensuring the existence of a unique optimal control.
• To demonstrate that the Atangana-Baleanu fractional derivative enables well-posedness for the full range of α ∈ (0,1), unlike classical fractional derivatives.
• To provide a constructive framework for computing the optimal control through the adjoint state system and the optimality condition.

Proposed method

• Utilizes spectral theory in Hilbert spaces to analyze the fractional diffusion equation with the Atangana-Baleanu fractional derivative of order α ∈ (0,1).
• Employs integration by parts formula for the Atangana-Baleanu fractional derivative to derive the weak formulation of the problem.
• Applies the Lax-Milgram lemma to prove existence and uniqueness of the weak solution for the state equation.
• Constructs the optimal control problem with a cost functional combining state tracking error and control regularization via a Tikhonov-type term.
• Derives the adjoint state equation backward in time with terminal condition η(T) = 0, using the time-reversed Atangana-Baleanu derivative.
• Applies the Lagrange multiplier method to derive the first-order optimality system, leading to the control formula ū = −η/𝒩.

Experimental results

Research questions

• RQ1 Can the existence and uniqueness of solutions for the fractional diffusion equation with Atangana-Baleanu time derivative be established for all α ∈ (0,1), including α ≤ 1/2?
• RQ2 How can optimal control be formulated and solved for a diffusion equation driven by a fractional derivative with a nonsingular, nonlocal Mittag-Leffler kernel?
• RQ3 What is the structure of the optimality system for such a control problem, and does it guarantee a unique optimal control?
• RQ4 How does the choice of the Atangana-Baleanu derivative affect the regularity and solvability of the state and adjoint equations compared to Caputo or Riemann-Liouville derivatives?
• RQ5 Can the optimal control be explicitly characterized in terms of the adjoint state, and what is the resulting feedback law?

Key findings

• The existence and uniqueness of the solution to the fractional diffusion equation with Atangana-Baleanu time derivative are proven for all α ∈ (0,1), extending previous results limited to α ∈ (1/2,1).
• The optimal control problem admits a unique solution, and the optimal control is characterized by the formula ū = −η/𝒩, where η is the solution of the adjoint equation.
• The first-order optimality system consists of the state equation, the adjoint equation with terminal condition, and the control formula, forming a complete characterization of the optimal control.
• The adjoint state η belongs to L²((0,T); H²(Ω) ∩ H₀¹(Ω)), ensuring sufficient regularity for the optimality condition to hold.
• The functional derivative of the cost functional vanishes at the optimal control, confirming that the derived control satisfies the necessary optimality condition.
• The method successfully generalizes optimal control theory to fractional PDEs with nonsingular, nonlocal kernels, offering a robust framework for complex systems with memory effects.