In this paper we consider a penalized Stokes equation defined in a regular domain Ω ⊂ R 2 and with Dirichlet boundary conditions. We prove that our system is null controllable using a scalar control defined in an open subset inside Ω and whose cost is bounded uniformly with respect to the parameter that converges to 0.
In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on [Formula: see text]. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman, we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in [Formula: see text] have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.
In this paper we consider a Stokes system with Navier-slip boundary conditions. The main results concern the behaviour of the cost of null controllability with respect to the diffusion coefficient when the control acts in the interior. In particular, we prove in (0, π) 2 that for a sufficiently large time the cost decays exponentially as the diffusion coefficient vanishes, whereas in (0, π) 3 we prove that for most of the control domains and for any time T > 0 the cost explodes exponentially as the diffusion coefficient vanishes.
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