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

Kévin Le Balc'h, Luz de Teresa

TL;DR

This work identifies a Kalman-type rank condition as the exact criterion for small-time null-controllability of a class of coupled Stokes systems with two constant coupling matrices, $D$ and $Q$, and distributed controls acting on distinct subdomains. When the condition holds, the authors derive a sharp control-cost bound of the form $\|v\|_{L^2(0,T;\mathcal U)} \le C e^{C/T} \|y_0\|_{\mathcal H^n}$ for any $T>0$, by fusing a spectral estimate for the Stokes operator with a Lebeau-Robbiano strategy and internal observability for low frequencies. The necessity part reduces to a finite-dimensional Kalman rank condition per Stokes eigenvalue, while sufficiency is established via spectral inequalities and a careful frequency decomposition, extending prior results to non-diagonal $D$ and separate control supports. The results advance the understanding of controllability criteria for coupled parabolic systems and provide an optimal-cost framework applicable to multi-physics fluid models.

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.

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

TL;DR

This work identifies a Kalman-type rank condition as the exact criterion for small-time null-controllability of a class of coupled Stokes systems with two constant coupling matrices, and , and distributed controls acting on distinct subdomains. When the condition holds, the authors derive a sharp control-cost bound of the form for any , by fusing a spectral estimate for the Stokes operator with a Lebeau-Robbiano strategy and internal observability for low frequencies. The necessity part reduces to a finite-dimensional Kalman rank condition per Stokes eigenvalue, while sufficiency is established via spectral inequalities and a careful frequency decomposition, extending prior results to non-diagonal and separate control supports. The results advance the understanding of controllability criteria for coupled parabolic systems and provide an optimal-cost framework applicable to multi-physics fluid models.

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.

Paper Structure

This paper contains 5 sections, 9 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Necessity of the Kalman rank condition
  3. Sufficiency of the Kalman rank condition and optimal cost
  4. Well-posedness of the coupled Stokes system
  5. A direct Lebeau-Robbiano strategy

Key Result

Theorem 1.2

System eq:controlform is null-controllable at time $T>0$ if and only if Moreover, if eq:kalman holds, then there exists $C=C(\Omega,\omega_1, \dots, \omega_m,D,Q,R)>0$ such that for every $T>0$, for any $y_0 \in \mathcal{H}^n$, there exists a control $v \in L^2(0,T;\mathcal{U})$ satisfying such that the corresponding solution $y$ of eq:controlform satisfies $y(T, \cdot) = 0$.

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • ...and 6 more