The Kalman rank condition and the optimal cost for the null-controllability of coupled Stokes systems

TL;DR

This paper establishes that coupled Stokes systems are null-controllable in small time if and only if a Kalman rank condition is met, and provides the sharp upper bound for the control cost.

Contribution

It introduces a Kalman rank criterion for null-controllability of coupled Stokes systems and derives the optimal control cost bounds.

Findings

Null-controllability holds if and only if the Kalman rank condition is satisfied.

The paper provides the sharp upper bound for the control cost.

It adapts spectral estimates and observability strategies to coupled Stokes systems.

Abstract

This paper considers the controllability of a class of coupled Stokes systems with distributed controls. The coupling terms are of a different nature. The first coupling is through the principal part of the Stokes operator with a constant real-valued positive-definite matrix. The second one acts through zero-order terms with a constant real-valued matrix. We assume the controls have their support in different measurable subsets of the spatial domain. Our main result states that such a system is small-time null-controllable if and only if a Kalman rank condition is satisfied. Moreover, when this condition holds, we prove the sharp upper bound for the cost of null-controllability for these systems. Our method is based on two ingredients. We start from the recent spectral estimate for the Stokes operator from Chaves-Silva, Souza, and Zhang. Then, we adapt Lissy and Zuazua's strategy concerning the internal observability for coupled systems of linear parabolic equations to coupled Stokes systems.