Electronic ISBN:  9781470404567 
Product Code:  MEMO/181/852.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 181; 2006; 128 ppMSC: Primary 76; 35;
The steadystate solutions to NavierStokes equations on a bounded domain \(\Omega \subset R^d\), \(d = 2,3\), are locally exponentially stabilizable by a boundary closedloop 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 nonlinearity 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 finitedimensional unstable subspace.
In order to inject dissipation as to force local exponential stabilization of the steadystate solutions, an Optimal Control Problem (OCP) with a quadratic cost functional over an infinite timehorizon is introduced for the linearized NS equations. As a result, the same Riccatibased, optimal boundary feedback controller which is obtained in the linearized OCP is then selected and implemented also on the full NS 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 freedynamics operator. In contrast, established (and rich) optimal control theory [LT.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 nonstandard 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 nonempty; that is, it satisfies the socalled Finite Cost Condition [LT.2]. 
Table of Contents

Chapters

1. Introduction

2. Main results

3. Proof of Theorems 2.1 and 2.2 on the linearized system (2.4): $d$ = 3

4. Boundary feedback uniform stabilization of the linearized system (3.1.4) via an optimal control problem and corresponding Riccati theory. Case $d$ = 3

5. Theorem 2.3(i): Wellposedness of the NavierStokes equations with Riccatibased boundary feedback control. Case $d$ = 3

6. Theorem 2.3(ii): Local uniform stability of the NavierStokes equations with Riccatibased boundary feedback control

7. A PDEinterpretation of the abstract results in Sections 5 and 6


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The steadystate solutions to NavierStokes equations on a bounded domain \(\Omega \subset R^d\), \(d = 2,3\), are locally exponentially stabilizable by a boundary closedloop 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 nonlinearity 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 finitedimensional unstable subspace.
In order to inject dissipation as to force local exponential stabilization of the steadystate solutions, an Optimal Control Problem (OCP) with a quadratic cost functional over an infinite timehorizon is introduced for the linearized NS equations. As a result, the same Riccatibased, optimal boundary feedback controller which is obtained in the linearized OCP is then selected and implemented also on the full NS 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 freedynamics operator. In contrast, established (and rich) optimal control theory [LT.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 nonstandard 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 nonempty; that is, it satisfies the socalled Finite Cost Condition [LT.2].

Chapters

1. Introduction

2. Main results

3. Proof of Theorems 2.1 and 2.2 on the linearized system (2.4): $d$ = 3

4. Boundary feedback uniform stabilization of the linearized system (3.1.4) via an optimal control problem and corresponding Riccati theory. Case $d$ = 3

5. Theorem 2.3(i): Wellposedness of the NavierStokes equations with Riccatibased boundary feedback control. Case $d$ = 3

6. Theorem 2.3(ii): Local uniform stability of the NavierStokes equations with Riccatibased boundary feedback control

7. A PDEinterpretation of the abstract results in Sections 5 and 6