# Tangential Boundary Stabilization of Navier-Stokes Equations

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*Viorel Barbu; Irena Lasiecka; Roberto Triggiani*

The steady-state solutions to Navier-Stokes equations on a
bounded domain \(\Omega \subset R^d\), \(d = 2,3\), are locally
exponentially stabilizable by a boundary closed-loop feedback controller,
acting tangentially on the boundary \(\partial \Omega\), 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
\((H^{\tfrac{3}{2}+\epsilon}(\Omega))^3\), \(\epsilon > 0\), a
high topological level. A first implication thereof is that, due to
compatibility conditions that now come into play, for \(d=3\), the
boundary feedback stabilizing controller must be infinite
dimensional. Moreover, it generally acts on the entire boundary \(\partial
\Omega\). Instead, for \(d=2\), where the topological level for
stabilization is \((H^{\tfrac{3}{2}-\epsilon}(\Omega))^2\), 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 equations. 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 controls, 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
\(\tfrac{3}{2}\), 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].

#### Table of Contents

# Table of Contents

## Tangential Boundary Stabilization of Navier-Stokes Equations

- Contents v6 free
- Acknowledgements ix10 free
- Chapter 1. Introduction 112 free
- Chapter 2. Main results 1324 free
- Chapter 3. Proof of Theorems 2.1 and 2.2 on the linearized system ( 2.4): d = 3 2132
- 3.1. Abstract models of the linearized problem ( 2.3). Regularity 2132
- 3.2. The operator D*A, D*:H→(L[sup(2)](T))[sub(D)] 3344
- 3.3. A critical boundary property related to the boundary c.c. in ( 3.1.2e) 3546
- 3.4. Some technical preliminaries; space and system decomposition 3647
- 3.5. Theorem 2.1, general case d = 3: An infinite-dimensional open–loop boundary controller g satisfying the FCC (3.1.22)–(3.1.24) for the linearized system… 3849
- 3.6. Feedback stabilization of the unstable [sub(Z)]N–system ( 3.4.9) on Z[sup(u)][sub(N)] under the FDSA 5061
- 3.7. Theorem 2.2, case d = 3 under the FDSA: An open–loop boundary controller g satisfying the FCC ( 3.1.22)–( 3.1.24) for the linearized system… 5768

- 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 6172
- 4.0. Orientation 6172
- 4.1. The optimal control problem ( Case d = 3) 6273
- 4.2. Optimal feedback dynamics: the feedback semigroup and its generator on W 6475
- 4.3. Feedback synthesis via the Riccati operator 6778
- 4.4. Identification of the Riccati operator R in ( 4.1.8) with the operator R[sub(1)] in ( 4.3.1) 7687
- 4.5. A Riccati–type algebraic equation satisfied by the operator R on the domain D(A[sup2)][Sub(R)], Where A[sub(R)] is the feedback generator 8091

- Chapter 5. Theorem 2.3(i): Well–posedness of the Navier–Stokes equations with Riccati–based boundary feedback control. Case d = 3 8394
- Chapter 6. Theorem 2.3(ii): Local uniform stability of the Navier–Stokes equations with Riccati–based boundary feedback control 93104
- Chapter 7. A PDE–interpretation of the abstract results in Sections 5 and 6 95106
- Appendix A. Technical Material Complementing Section 3.1 97108
- Appendix B. Boundary feedback stabilization with arbitrarily small supportof the linearized system 103114
- B.1. An open…loop infinite dimensional boundary controller g ε L[sup(2)](0,∞); (L[sup(2)](T[sub(1)])[sup[sup(d)]), T[sub(1)] arbitrary, for the linearized system 103114
- B.2. Feedback stabilization in (H[sup3/2…ε)(Ω))[sup(d)], d = 2,3, of the N…S linearized system 104115
- B.3. Completion of the proof of Theorem 2.5 and Theorem 2.6 for the N–S model (1.1), d = 2 106117
- B.4. A regularity property of the Riccati operator corresponding to the linearized operator A in (1.11) 107118

- Appendix C. Equivalence between unstable and stable versions of the Optimal Control Problem of Section 4 113124
- Appendix D. Proof that FS(.) εL(W;L[sup(2)](0,∞);(L[sup(2)](T))[sup(d)] 123134
- Bibliography 127138