Tangential boundary stabilization of navier -stokes equations
The steady-state solutions to Navier-Stokes equations on a bounded domain Ω ⊂ Rd, d = 2, 3, are locally exponentially stabilizable by a boundary closed-loop feedback controller, acting tangentially on the boundary ∂Ω, 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 2/3+c(Ω))3, ∈ > 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 ∂Ω Instead, for d = 2, where the topological level for stabilization is (H3/2-ε(Ω))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 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].
Memoirs of the American Mathematical Society
Barbu, V., Lasiecka, I., & Triggiani, R. (2006). Tangential boundary stabilization of navier -stokes equations. Memoirs of the American Mathematical Society, 181 (852), 1-136. https://doi.org/10.1090/memo/0852