Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint
Provides rigorous numerical analysis for optimal control of subdiffusion processes, which is important for applications in anomalous diffusion but the results are incremental as they extend existing techniques to a specific fractional PDE setting.
The paper establishes pointwise-in-time error estimates for a distributed optimal control problem governed by a subdiffusion equation with fractional time derivative, achieving convergence rates of O(τ^{min(1/2+α-ε,1)}+h^2) in discrete L^2 norm and O(τ^{α-ε}+ℓ_h^2 h^2) in discrete L^∞ norm.
In this work, we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation which involves a fractional derivative of order $α\in(0,1)$ in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by a variational type discretization. With a space mesh size $h$ and time stepsize $τ$, we establish the following order of convergence for the numerical solutions of the optimal control problem: $O(τ^{\min({1}/{2}+α-ε,1)}+h^2)$ in the discrete $L^2(0,T;L^2(Ω))$ norm and $O(τ^{α-ε}+\ell_h^2h^2)$ in the discrete $L^\infty(0,T;L^2(Ω))$ norm, with any small $ε>0$ and $\ell_h=\ln(2+1/h)$. The analysis relies essentially on the maximal $L^p$-regularity and its discrete analogue for the subdiffusion problem. Numerical experiments are provided to support the theoretical results.