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Tangential boundary stabilization of Navier-Stokes equations
ArticleinMemoirs of the American Mathematical Society · May 2006
DOI: 10.1090/memo/0852
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Viorel Barbu
Universitatea Alexandru Ioan Cuza
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Irena Lasiecka
University of Virginia
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Roberto Triggiani
The University of Memphis
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Boundary Stabilization of Navier-Stokes Equations
Viorel Barbu
Irena Lasiecka
Roberto Triggiani
Author address:
Department of Mathematics, University of Virginia, Charlottesville,
VA 22904

Contents
Acknowledgements 1
Chapter 1. Introduction 3
Chapter 2. Main results 15
Chapter 3. Proof of Theorems 2.1 and 2.2 on the linearized system (2.4):
d=3 21
3.1. Abstract models of the linearized problem (2.3). Regularity 21
3.2. The operator D∗A∗,D∗:H→(L2(Γ))d33
3.3. A critical boundary property related to the boundary c.c.
in (3.1.2e) 35
3.4. Some technical preliminaries; space and system decomposition 36
3.5. Theorem 2.1, general case d= 3: An infinite-dimensional open-loop
boundary controller gsatisfying the FCC (3.1.22)–(3.1.24) for the
linearized system (3.1.4): g∈L2(0,∞;(L2(Γ))3), g·ν≡0 on Σ;
y0∈(H1
2+(Ω))3∩H⇒y∈L2(0,∞;(H3 2+(Ω))3∩H)37
3.6. Feedback stabilization of the unstable zN-system (3.4.9) on Zu
N
under the FDSA 49
3.7. Theorem 2.2, case d= 3 under the FDSA: An open-loop boundary
controller gsatisfying the FCC (3.1.22)–(3.1.24) for the linearized
system (3.1.4): g∈L2(0,∞;(L2(Γ1))d), g·ν= 0 on Σ, g≡0on
Γ\Γ1;y0∈(H1
2+(Ω))d∩H⇒y∈L2(0,∞;(H3 2+2(Ω))d∩H)56
Chapter 4. Boundary feedback uniform stabilization of the linearized system
(3.1.4) via an optimal control problem and corresponding
Riccati theory. Case d=3 59
4.1. The optimal control problem (Case d= 3) 60
4.2. Optimal feedback dynamics: the feedback semigroup and its
generator on W62
4.3. Feedback synthesis via the Riccati operator 65
4.4. Identification of the Riccati operator Rin (4.1.8) with the
operator R1in (4.3.1) 74
4.5. A Riccati-type algebraic equation satisfied by the operator R
on the domain D(A2
R), where ARis the feedback generator 78
Chapter 5. Theorem 2.3(i): Well-posedness of the Navier-Stokes equations
with Riccati-based boundary feedback control. Case d=3 81
Chapter 6. Theorem 2.3(ii): Local uniform stability of the Navier-Stokes
equations with Riccati-based boundary feedback control 91
5

6 CONTENTS
Chapter 7. A PDE-interpretation of the abstract results in Sections 5 and 6 93
Appendix A. Technical Material Complementing Section 3.1 95
A.1. Extension of the Leray Projector POutside the Space (L2(Ω))d95
Definition A.1.2: Extension of Leray projector to H−1(Ω))d.96
A.2. Definition and Regularity of the Dirichlet Map in the General Case.
Abstract Model 97
Appendix B. Boundary feedback stabilization with arbitrarily small support
of the linearized system (3.1.4a) at the (H3
2−(Ω))d∩H-level,
with I.C. y0∈(H1
2−(Ω))d∩H. Cases d=2,3 101
B.1. Theorem 2.5, general case d= 2: An open-loop infinite dimensional
boundary controller g∈L2(0,∞;(L2(Γ1))d), Γ1arbitrary, for the
linearized system (3.1.4a) yielding y(·;y0)∈L2(0,∞;(H3
2−(Ω))d)
for y0∈(H1
2−(Ω))d101
B.2. Feedback stabilization in (H3
2−(Ω))d,d=2,3, of the linearized
system (3.1.4a) with finite-dimensional feedback controller with
arbitrary support under FDSA = (3.6.2) 102
B.3. Completion of the proof of Theorem 2.5 and Theorem 2.6
for the N-S model (1.1), d= 2 104
Appendix C. Equivalence between unstable and stable versions of the
Optimal Control Problem of Section 4 105
Appendix D. Proof that FS(·)∈L(W;L2(0,∞;(L2(Γ))d) 115
Bibliography 119

Abstract
The steady-state solutions to Navier-Stokes equations on a bounded domain
Ω⊂Rd,d=2,3, are locally exponentially stabilizable by a boundary closed-
loop feedback controller, acting on the boundary ∂Ω, in the Dirichlet boundary
conditions. The greatest challenge arises from a combination between the control
as acting on the boundary and the dimensionality d= 3. If d= 3, the non-linearity
imposes and dictates the requirement that stabilization must occur in the space
(H3
2+(Ω))3,>0, a high topological level. A first implication thereof is that, for
d= 3, the boundary feedback stabilizing controller must be infinite dimensional.
Moreover, it generally acts on the entire boundary ∂Ω. Instead, for d= 2, where the
topological level for stabilization is (H3
2−(Ω))2, the boundary feedback stabilizing
controller can be chosen to act on an arbitrarily small portion of the boundary.
Moreover, still for d= 2, it may even be finite dimensional, and this occurs if the
linearized operator is diagonalizable over its finite-dimensional unstable subspace.
In order to inject dissipation as to force local exponential stabilization of the
steady-state solutions, an Optimal Control Problem (OCP) with a quadratic cost
functional over an infinite time-horizon is introduced for the linearized N-S equa-
tions. As a result, the same Riccati-based, optimal boundary feedback controller
which is obtained in the linearized OCP is then selected and implemented also on
the full N-S system. For d= 3, the OCP falls definitely outside the boundaries
of established optimal control theory for parabolic systems with boundary con-
trols, in that the combined index of unboundedness—between the unboundedness
of the boundary control operator and the unboundedness of the penalization or
observation operator—is strictly larger than 3
2, as expressed in terms of fractional
powers of the free-dynamics operator. In contrast, established (and rich) optimal
control theory [L-T.2] of boundary control parabolic problems and corresponding
algebraic Riccati theory requires a combined index of unboundedness strictly less
than 1. An additional preliminary serious difficulty to overcome lies at the outset of
the program, in establishing that the present highly non-standard OCP—with the
aforementioned high level of unboundedness in control and observation operators
and subject, moreover, to the additional constraint that the controllers be pointwise
tangential—be non-empty; that is, it satisfies the so-called Finite Cost Condition
[L-T.2].
2000 Mathematics Subject Classification. Primary: 76D05, 7655, 35B40, 35Q30.
Key words and phrases. Navier-Stokes equations, boundary feedback stabilization, optimal
control, Riccati equation, steady-state solution, boundary feedback controller.
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