No abstract

In this paper we deal with a robust Stackelberg strategy for the Navier-Stokes system. The scheme is based in considering a robust control problem for the "follower control" and its associated disturbance function. Afterwards, we consider the notion of Stackelberg optimization (which is associated to the "leader control") in order to deduce a local null controllability result for the Navier-Stokes system.

This paper studies the internal control of the Korteweg-de Vries-Burgers (KdVB) equation on a bounded domain. The diffusion coefficient is time-dependent and the boundary conditions are mixed in the sense that homogeneous Dirichlet and periodic Neumann boundary conditions are considered. The exact controllability to the trajectories is proven for a linearized system by using duality and getting a new Carleman estimate. Then, using an inversion theorem we deduce the local exact controllability to the trajectories for the original KdVB equation, which is nonlinear.

In this paper, we prove Lipschitz stability results for the inverse source problem of determining the spatially varying factor in a source term in the Korteweg–de Vries–Burgers (KdVB) equation with mixed boundary conditions. More precisely, the Lipschitz stability property is obtained using observation data on an arbitrary fixed sub-domain over a time interval. Secondly, we show that stability property can also be achieved from boundary measurements. Our proofs relies on Carleman inequalities and the Bukhgeim–Klibanov method.

We consider the inverse problem of determining the spatial dependence of a source of the form f (x) sigma (t) in the Stokes system defined in Omega x (0, T), assuming that sigma (t) is known and f (x) is divergence-free. The only available observation is a single internal measurement of the velocity and the acceleration, for which one of its components is missing. Under adequate hypothesis on sigma we prove uniqueness of this inverse problem and we establish an explicit reconstruction formula. This formula provides the spectral coefficients f(k) of the source f in terms of a family of null controls h((tau)) for the corresponding adjoint system indexed by tau is an element of (0, T].