Optimal control of variable-exponent subdiffusion

TL;DR

This paper develops an optimal control framework for variable-exponent subdiffusion models, extending previous work to account for multiscale diffusion behavior, and provides theoretical analysis and numerical validation of the proposed methods.

Contribution

It introduces a convolution reformulation for variable-exponent subdiffusion control problems and establishes well-posedness, regularity, and numerical accuracy results.

Findings

Proved well-posedness and regularity of the control problem.

Established $O( au(1+| ext{ln} au|)+h^2)$ accuracy of numerical schemes.

Numerical experiments confirm theoretical results.

Abstract

This work investigates the optimal control of the variable-exponent subdiffusion, which extends the work [Gunzburger and Wang, {\it SIAM J. Control Optim.} 2019] to the variable-exponent case to account for the multiscale and crossover diffusion behavior. To resolve the difficulties caused by the leading variable-exponent operator, we adopt the convolution method to reformulate the model into an equivalent but more tractable form, and then prove the well-posedness and weighted regularity of the optimal control. As the convolution kernels in reformulated models are indefinite-sign, non-positive-definite, and non-monotonic, we adopt the discrete convolution kernel approach in numerical analysis to show the $O(\tau(1+|\ln\tau|)+h^2)$ accuracy of the schemes for state and adjoint equations. Numerical experiments are performed to substantiate the theoretical findings.