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

TL;DR

This paper analyzes the null controllability cost of the n-dimensional Stokes system with controls on n-1 components, showing the cost remains unchanged compared to controls on all n components, through spectral estimates.

Contribution

It introduces a spectral estimate for low frequencies of the Stokes operator with n-1 controls, demonstrating the controllability cost is unaffected by removing one control component.

Findings

Controllability cost is of order $O(e^{C/T})$ with n-1 controls.

Spectral estimate for low frequencies involving only n-1 components.

Null controllability cost remains unchanged despite control reduction.

Abstract

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.