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Single drift control controllability for dynamic-boundary SPDEs

Establish null or approximate controllability at time T for the forward stochastic parabolic equation with dynamic boundary conditions when only a single localized control u acts on the drift term in the interior (i.e., v1 ≡ 0 and v2 ≡ 0), by proving the required observability inequality or unique continuation property for the adjoint system based solely on localized measurements of the bulk state.

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Background

In the main controllability results, the authors employ controls not only in the interior drift (u) but also in the diffusion terms (v1 and v2) to derive observability inequalities and unique continuation properties for the adjoint system.

They explicitly state that removing the diffusion controls v1 and v2 and relying solely on the localized drift control u leads to a challenging open problem, as one must prove a stronger observability inequality involving only the localized bulk state or an equivalent unique continuation property.

Partial results exist in special cases, but a general theory for this single-control configuration under dynamic boundary conditions has not been established.

References

The null (or approximate) controllability of equation 1.4, with only the control u acting on the drift term of the bulk equation, is a challenging open problem.

1.4:

$\begin{cases} \begin{array}{ll} dy - \nabla\cdot(\mathcal{A}\nabla y) \,dt = (a_1 y + B_1 \cdot \nabla y + \mathbbm{1}_{G_0} u) \,dt + v_1 \,dW(t) &\text{in}\,\, Q,\\ dy_\Gamma - \nabla_\Gamma \cdot (\mathcal{A}_\Gamma \nabla_\Gamma y_\Gamma) \,dt + \partial_\nu^\mathcal{A} y \,dt = (a_2 y_\Gamma + B_2 \cdot \nabla_\Gamma y_\Gamma) \,dt + v_2 \,dW(t) &\text{on}\,\, \Sigma,\\ y_\Gamma = y|_\Gamma &\text{on}\,\, \Sigma,\\ (y, y_\Gamma)|_{t=0} = (y_0, y_{\Gamma,0}) &\text{in}\,\, G \times \Gamma, \end{array} \end{cases} $

— Carleman Estimates and Controllability of Forward Stochastic Parabolic Equations with General Dynamic Boundary Conditions (2510.12345 - Boulite et al., 14 Oct 2025) in Remark, Section 1 (Introduction and Main Results)