Tangential Boundary Stabilization of Navier-Stokes Equations.

Intro -- Contents -- Acknowledgements -- Chapter 1. Introduction -- Chapter 2. Main results -- Chapter 3. Proof of Theorems 2.1 and 2.2 on the linearized system ( 2.4): d = 3 -- 3.1. Abstract models of the linearized problem ( 2.3). Regularity -- 3.2. The operator D*A, D*:H→(L[sup(2)](T))[sub(D)] -- 3.3. A critical boundary property related to the boundary c.c. in ( 3.1.2e) -- 3.4. Some technical preliminaries -- space and system decomposition -- 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… -- 3.6. Feedback stabilization of the unstable [sub(Z)]N-system ( 3.4.9) on Z[sup(u)][sub(N)] under the FDSA -- 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… -- 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 -- 4.0. Orientation -- 4.1. The optimal control problem ( Case d = 3) -- 4.2. Optimal feedback dynamics: the feedback semigroup and its generator on W -- 4.3. Feedback synthesis via the Riccati operator -- 4.4. Identification of the Riccati operator R in ( 4.1.8) with the operator R[sub(1)] in ( 4.3.1) -- 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 -- Chapter 5. Theorem 2.3(i): Well-posedness of the Navier-Stokes equations with Riccati-based boundary feedback control. Case d = 3 -- Chapter 6. Theorem 2.3(ii): Local uniform stability of the Navier-Stokes equations with Riccati-based boundary feedback control -- Chapter 7. A PDE-interpretation of the abstract results in Sections 5 and 6.

Appendix A. Technical Material Complementing Section 3.1 -- A. l. Extension of the Leray Projector P Outside the Space (L[sup(2)](Ω))[sup(d)] -- A. 2. Definition and Regularity of the Dirichlet Map in the General Case. Abstract Model -- Appendix B. Boundary feedback stabilization with arbitrarily small supportof the linearized system -- 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 -- B.2. Feedback stabilization in (H[sup3/2…ε)(Ω))[sup(d)], d = 2,3, of the N…S linearized system -- B.3. Completion of the proof of Theorem 2.5 and Theorem 2.6 for the N-S model (1.1), d = 2 -- B.4. A regularity property of the Riccati operator corresponding to the linearized operator A in (1.11) -- Appendix C. Equivalence between unstable and stable versions of the Optimal Control Problem of Section 4 -- Appendix D. Proof that FS(.) εL(W -- L[sup(2)](0,∞) -- (L[sup(2)](T))[sup(d)] -- Bibliography.

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