Gradient Mittag-Leffler and strong stabilizability of time fractional diffusion processes

This paper deals with the gradient stability and the gradient stabilizability of Caputo time fractional diffusion linear systems. First, we give sufficient conditions that allow the gradient Mittag-Leffler and strong stability, where we use a direct method based essentially on the spectral properties of the system dynamic. Moreover, we consider a class of linear and distributed feedback controls that Mittag-Leffler and strongly stabilize the state gradient. The proposed results lead to an algorithm that allows us to gradient stabilize the state of the fractional systems under consideration. Finally, we illustrate the effectiveness of the developed algorithm by a numerical example and simulations.

💡 Research Summary

The paper investigates gradient stability and gradient stabilizability for linear diffusion systems governed by Caputo time‑fractional derivatives of order q∈(0,1]. The authors consider the abstract evolution equation
 C Dₜ^{q} y(x,t)=A y(x,t)+L v(x,t), y|_{∂Ω}=0, y(x,0)=y₀(x),
where A is a uniformly elliptic operator generating a C₀‑semigroup on Y=H¹(Ω), L is a bounded control operator, and the gradient operator ∇:Y→L²(Ω)ⁿ is the spatial derivative. Two notions of gradient stability are introduced: (i) gradient Mittag‑Leffler stability, meaning that the gradient norm decays like a power of the Mittag‑Leffler function E_q(−ξ t^{q}), and (ii) gradient strong stability, meaning the gradient norm tends to zero as t→∞.

The first main result (Theorem 1) provides spectral sufficient conditions for these properties. By decomposing the spectrum of A into the subsets ω₁(A) (non‑negative eigenvalues whose eigenfunctions lie in the kernel of ∇*∇) and ω₂(A) (negative eigenvalues with the same property), the theorem states that if ω₁(A) is empty and every λ∈ω₂(A) satisfies λ≤−ξ for some ξ>0, then the uncontrolled system is both gradient Mittag‑Leffler stable and gradient strongly stable. The proof relies on the eigenfunction expansion of the solution, the monotonicity of E_q(−t^{q}) for q∈(0,1), and the bound E_q(−ξ t^{q})≤C t^{−q}.

Next, the paper defines gradient stabilizability: a system is gradient Mittag‑Leffler (or strongly) stabilizable if there exists a bounded feedback operator D such that the closed‑loop system C Dₜ^{q} y=(A+L D) y enjoys the corresponding gradient stability. Two constructive design approaches are presented.

1. Decomposition Method – Assuming A is self‑adjoint with compact resolvent, its spectrum is split into a finite set of non‑negative eigenvalues ω_u and the remaining negative part ω_s, separated by a gap β>0. The projection P onto the span of the eigenfunctions associated with ω_u yields two subsystems: a finite‑dimensional “unstable” part (10) and an infinite‑dimensional “stable” part (11). By choosing a feedback D_u acting only on the unstable subspace so that the finite‑dimensional subsystem becomes gradient Mittag‑Leffler stable, and exploiting the intrinsic decay of the stable part, the whole system inherits gradient Mittag‑Leffler stability. Explicit estimates (13)–(18) show how the gradient norm of the full state is bounded by a combination of the two subsystems’ bounds.

2. Spectral Shift Method – Here the feedback D is selected so that the operator A+L D itself satisfies the spectral conditions of Theorem 1 (i.e., ω₁(A+L D)=∅ and all eigenvalues ≤−ξ). In this case the simple feedback law v(t)=D y(t) directly yields gradient Mittag‑Leffler (and consequently strong) stabilizability (Theorem 3). The proof is a straightforward adaptation of Theorem 1’s argument.

Based on these theoretical results, the authors propose an algorithm: given an initial state, a tolerance ε, and the fractional order q, iteratively apply the feedback law, compute the solution using the Mittag‑Leffler series, evaluate the gradient, and stop when the gradient norm falls below ε.

A numerical illustration is provided for a one‑dimensional domain Ω=(0,1) with A=∂²_x+π², L=π I, and D=−π I, yielding the feedback v(t)=−π² y(t). The eigenvalues λ_n=−n²π²+π² are all negative, satisfying the spectral gap condition. Simulations confirm that the gradient norm decays according to the Mittag‑Leffler profile and eventually becomes negligible, validating the theoretical predictions.

In conclusion, the paper extends the stability theory of fractional diffusion equations from the state level to the gradient level, offering spectral‑based sufficient conditions and two practically implementable feedback designs. The results are relevant for applications where control of spatial derivatives—such as heat flux, stress gradients, or concentration gradients—is essential. The authors suggest future work on nonlinear extensions, regional (sub‑domain) gradient control, and optimal control formulations.