On the Controllability of a Fully Nonlocal Coupled Stochastic Reaction--Convection--Diffusion System

In this paper, we study the null and approximate controllability of a class of fully nonlocal coupled stochastic reaction–convection–diffusion systems. The system consists of two forward stochastic parabolic equations driven by general second-order differential operators and incorporates four nonlocal zero-order integral terms. The nonlocality arises from integral kernel terms present in both equations, defined over a bounded domain $G \subset \mathbb{R}^N$ ($N \geq 1$). Since the coefficients depend on time, space, and random variables, we introduce three controls: a spatially localized control acting on the drift term of the first equation, and two additional controls acting on the diffusion terms of both equations. These additional controls are necessary to overcome difficulties due to the stochastic nature of the associated adjoint backward system. Using a standard duality argument, the controllability problem for the forward system is reduced to an observability problem for the corresponding adjoint nonlocal backward system. To establish this observability, we derive a new global Carleman estimate for the adjoint system, in which the drift terms belong to a negative Sobolev space and the equations include nonlocal integral terms. Our results are obtained under suitable cascade structure conditions on the coupling zero-order, nonlocal, and first-order terms of the system.

💡 Research Summary

This paper investigates null and approximate controllability for a class of fully non‑local coupled stochastic reaction–convection–diffusion systems. The forward system consists of two stochastic parabolic equations driven by general second‑order differential operators L₁(t) and L₂(t). Both equations contain zero‑order reaction terms a_{ij}, first‑order convection terms B_{ij}·∇, and four non‑local integral terms A₁, A₂ defined through kernels K_{ij}(t,x,β,ω). The kernels are assumed to be uniformly bounded in a weighted L²‑norm (condition (1.1)) that reflects the Carleman weight exp(−σ₀ t(T−t)).

Three controls are introduced: a spatially localized control f acting on the drift of the first equation (through the characteristic function χ_{G₀}), and two diffusion controls g₁, g₂ acting on the stochastic terms of the first and second equations, respectively. The diffusion controls are essential to compensate the stochastic nature of the adjoint backward system.

The authors reduce the controllability problem to an observability inequality for the adjoint backward system by a standard duality argument. The adjoint system is also coupled, contains the adjoint non‑local operators A₁* and A₂*, and features the same stochastic noises. The main technical contribution is a new global Carleman estimate for this backward system. Unlike classical Carleman estimates, the drift coefficients belong only to a negative Sobolev space H⁻¹, and the estimate must handle the integral kernels. By carefully designing the Carleman weight and exploiting the boundedness of the kernels, the authors obtain a weighted energy inequality that yields an observability estimate on the control region G₀.

To make the observability work, the paper imposes a cascade structure on the coupling terms: (i) the zero‑order coupling a_{21} is uniformly positive (or uniformly negative) on a small subdomain eG₀⊂G₀; (ii) the non‑local kernel K_{21} satisfies a smallness condition (1.9) with constants δ₀, δ₁; (iii) the first‑order coupling B_{21} is set to zero. These conditions guarantee that the localized control f influences the first component y directly and that its effect propagates to the second component z through the strong zero‑order and weak non‑local couplings, forming a “cascade”.

The paper proves two main results. Theorem 2.1 (null controllability) states that for any initial data (y₀,z₀)∈L²_F₀(Ω;L²(G;ℝ²)) and source terms (ξ₁,ξ₂) in a suitable subspace, there exist controls (f,g₁,g₂)∈L²_F(0,T;L²(G₀))×