Tangential boundary stabilization of Navier-Stokes equations
著者
書誌事項
Tangential boundary stabilization of Navier-Stokes equations
(Memoirs of the American Mathematical Society, no. 852)
American Mathematical Society, 2006
大学図書館所蔵 全13件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
  スウェーデン
  ノルウェー
  アメリカ
注記
"Volume 181, number 852 (first of 5 numbers)."
Bibliography: p. 127-128
内容説明・目次
内容説明
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].
目次
- Introduction Main results Proof of Theorems 2.1 and 2.2 on the linearized system (2.4): $d=3$ Boundary feedback uniform stabilization of the linearized system (3.1.4) via an optimal control problem and corresponding Riccati theory. Case $d=3$ Theorem 2.3(i): Well-posedness of the Navier-Stokes equations with Riccati-based boundary feedback control. Case $d=3$ Theorem 2.3(ii): Local uniform stability of the Navier-Stokes equations with Riccati-based boundary feedback control A PDE-interpretation of the abstract results in Sections 5 and 6 Appendix A. Technical material complementing Section 3.1 Appendix B. Boundary feedback stabilization with arbitrarily small support of the linearized system (3.1.4a) at the $(H^{\tfrac{3}{2}-\epsilon}(\Omega))^d \cap H$-level, with I.C. $y^0\in (H^{\frac{1}{2}-\epsilon}(\Omega))^d \cap H$. Cases $d=2,3$. Theorem 2.5 for $d=2$ Appendix C. Equivalence between unstable and stable versions of the optimal control problem of Section 4 Appendix D. Proof that $FS(\cdot) \in \mathcal {L}(W
- L^2(0,\infty
- (L^2(\Gamma))^d)$ Bibliography.
「Nielsen BookData」 より