Tangential Boundary Stabilization of Navier-Stokes Equations: Volume 181, Issue 852
Viorel Barbu · Irena Lasiecka · Roberto Triggiani
јан 2006. · American Mathematical Soc.
Е-књига
128
Страница
reportОцене и рецензије нису верификоване Сазнајте више
О овој е-књизи
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].
Оцените ову е-књигу
Јавите нам своје мишљење.
Информације о читању
Паметни телефони и таблети
Инсталирајте апликацију Google Play књиге за Android и iPad/iPhone. Аутоматски се синхронизује са налогом и омогућава вам да читате онлајн и офлајн где год да се налазите.
Лаптопови и рачунари
Можете да слушате аудио-књиге купљене на Google Play-у помоћу веб-прегледача на рачунару.
Е-читачи и други уређаји
Да бисте читали на уређајима које користе е-мастило, као што су Kobo е-читачи, треба да преузмете фајл и пренесете га на уређај. Пратите детаљна упутства из центра за помоћ да бисте пренели фајлове у подржане е-читаче.
