Controllability to trajectories of a Ladyzhenskaya model for a viscous incompressible fluid

Abstract

We consider the controllability of a viscous incompressible fluid modeled by the Navier-Stokes system with a nonlinear viscosity. To prove the controllability to trajectories, we linearize around a trajectory and the corresponding linear system includes a nonlocal spatial term. Our main result is a Carleman estimate for the adjoint of this linear system. This estimate yields in a standard way the null controllability of the linear system and the local controllability to trajectories. Our method to obtain the Carleman estimate is completely general and can be adapted to other parabolic systems when a Carleman estimate is available.

Reachability results for perturbed heat equations

“…Remark A.1. The sufficient condition for unique continuation stated in Proposition A.2 is in fact closely related to the one used in the recent work of Guerrero and Takahashi [15], see in particular conditions (1.10), (1.11) and Remark 1.2 therein.…”

Control Problems for the Navier–Stokes System with Nonlocal Spatial Terms

“…These properties allow us to deduce Corollary 1.2 from Theorem 1.1 (see, for instance, Section 4 in [16]) by again assuming λ ⩾ λ 0 and s ⩾ s 0 (T m + T 2m ) for some s 0 > 0 and λ 0 > 0.…”

“…The functions σ 1 , σ 2 , σ 3 are defined precisely by (3.1), (3.2) in Section 3 by using the Carleman weights that we describe in Section 2.2. By standard methods (see, for instance, Section 4 in [16]), we deduce from Theorem 1.1 the following result: Corollary 1.2. Assume (1.5), (1.7) and (1.8).…”

“…The control u is localized in a nonempty domain ω ⊂ Ω. A similar system was considered in [16] for controllability issues, and in that article, the nonlocal spatial term appears from the linearization of a model considered by Ladyzhenskaya [19]. Some authors have also considered the controllability of partial differential equations with nonlocal spatial terms in [2,13,14,17,18,20,21,22,24].…”

“…They obtain the local exact controllability to trajectories with a passage to the limit. In [16], we consider a kernel of the above form but with n = 1. Here we deal with the case n ⩾ 1 and we use a control with one vanishing component.…”