Optimal cost for the null controllability of the Stokes system with controls having $n-1$ components and applications

In this work, we investigate the optimal cost of null controllability for the $n$-dimensional Stokes system when the control acts on $n-1$ scalar components. We establish a novel spectral estimate for low frequencies of the Stokes operator, involving solely $n-1$ components, and use it to show that the cost of controllability with controls having $n-1$ components remains of the same order in time as in the case of controls with $n$ components, namely $O(e^{C/T})$, i.e. the cost of null controllability is not affected by the absence of one component of the control. We also give several applications of our results.

💡 Research Summary

The paper addresses the null‑controllability problem for the linear Stokes system in a bounded domain Ω⊂ℝⁿ (n=2,3) when the control acts only on n‑1 scalar components of the velocity field. Classical results guarantee null‑controllability when all n components are controlled, with an optimal cost of order e^{C/T}. However, when one component is omitted, previous works based on Carleman estimates only yielded sub‑optimal cost bounds (e^{C/T⁴} or e^{C/T⁹}) and often required geometric restrictions on the control region ω.

The authors prove that the optimal exponential cost remains unchanged even with n‑1 components. The main contributions are:

1. Spectral inequality for low frequencies (Theorem 1.5). For any cutoff Λ and any coefficient sequence {a_j}, the sum of squares of the coefficients is bounded by an exponential factor times the L²‑norm over ω of the linear combination of the first n‑1 components of the Stokes eigenfunctions. This inequality is proved by a semiclassical analysis that introduces a small parameter h≈Λ^{-1/2} and exploits the ellipticity of the Laplacian on the velocity field while carefully handling the pressure term.

2. Interpolation inequality (Theorem 1.6). Using the spectral inequality, the authors establish a quantitative estimate that interpolates between the H¹‑norm of a modified velocity field on a larger set Z and its H¹‑norm on a smaller set W, plus a weighted L²‑norm on the control region. The proof relies on pseudo‑differential calculus and Gårding’s inequality, and it avoids the need for Carleman weights.

3. Observability inequality (Theorem 1.4). Combining the interpolation and spectral inequalities yields an observability estimate for the adjoint Stokes system: the H‑norm of the solution at final time T is bounded by C₁e^{C₂/T} times the integral over (0,T)×ω of the squared magnitude of the first n‑1 components. This inequality is the dual of the controllability result and provides the exact exponential dependence on the control time.

4. Null‑controllability with n‑1 components (Theorem 1.3). By duality, the observability inequality translates into the existence of a control f∈L²(0,T;L²(ω)^{n‑1}) such that the velocity field reaches zero at time T, with the cost estimate
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