Controllability without surface diffusion in the dynamic boundary condition
Determine the null and approximate controllability at time T for the forward stochastic parabolic system on Q = (0, T) × G with dynamic boundary conditions when the boundary equation has no surface diffusion term, namely: dy − ∇·(𝒜∇y) dt = (a1 y + B1·∇y + 1_{G0} u) dt + v1 dW(t) in Q, dy_Γ + ∂_ν^𝒜 y dt = (a2 y_Γ + B2·∇_Γ y_Γ) dt + v2 dW(t) on Σ = (0, T) × Γ, and y_Γ = y|_Γ on Σ, where 𝒜 and 𝒜_Γ are uniformly elliptic diffusion matrices and a1, a2, B1, B2 are bounded adapted coefficients. Establish whether controls (u, v1, v2) can drive the state to zero (null controllability) or arbitrarily close to a target (approximate controllability) under this boundary configuration.
References
However, the study of the controllability of 1.4 with the following dynamic boundary condition:
\begin{equation*}
\begin{cases}
dy_\Gamma + \partial_\nu\mathcal{A} y \, dt = (a_2 y_\Gamma + B_2 \cdot \nabla_\Gamma y_\Gamma) \, dt + v_2 \, dW(t) & \textnormal{on} \; \Sigma, \
y_\Gamma = y|_\Gamma & \textnormal{on} \; \Sigma, \end{cases}
\end{equation*}
remains an 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} $