Tangential boundary stabilization of Navier-Stokes equations /
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| Author / Creator: | Barbu, Viorel. |
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| Imprint: | Providence, R.I. : American Mathematical Society, c2006. |
| Description: | ix, 128 p. ; 26 cm. |
| Language: | English |
| Series: | Memoirs of the American Mathematical Society, 0065-9266 ; no. 852 |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/6448044 |
| Summary: | 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]. |
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| Item Description: | "Volume 181, number 852 (first of 5 numbers)." |
| Physical Description: | ix, 128 p. ; 26 cm. |
| Bibliography: | Includes bibliographical references. |
| ISBN: | 0821838741 9780821838747 |
