Tangential Boundary Stabilization of Navier-Stokes Equations

Released on by Viorel Barbu
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  • Tangential Boundary Stabilization of Navier-Stokes Equations Book Detail

  • Author : Viorel Barbu
  • Release Date : 2006
  • Publisher : American Mathematical Soc.
  • Genre : Mathematics
  • Pages : 146
  • ISBN 13 : 0821838741
  • File Size : 37,37 MB
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Tangential Boundary Stabilization of Navier-Stokes Equations by Viorel Barbu PDF Summary

Book Description: 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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