Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing

We study controllability issues for the 2D Euler and Navier- Stokes (NS) systems under periodic boundary conditions. These systems describe motion of homogeneous ideal or viscous incompressible fluid on a two-dimensional torus T^2. We assume the system to be controlled by a degenerate forcing applied to fixed number of modes. In our previous work [3, 5, 4] we studied global controllability by means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing viscosity (\nu > 0). Methods of dfferential geometric/Lie algebraic control theory have been used for that study. In [3] criteria for global controllability of nite-dimensional Galerkin approximations of 2D and 3D NS systems have been established. It is almost immediate to see that these criteria are also valid for the Galerkin approximations of the Euler systems. In [5, 4] we established a much more intricate suf- cient criteria for global controllability in finite-dimensional observed component and for L2-approximate controllability for 2D NS system. The justication of these criteria was based on a Lyapunov-Schmidt reduction to a finite-dimensional system. Possibility of such a reduction rested upon the dissipativity of NS system, and hence the previous approach can not be adapted for Euler system. In the present contribution we improve and extend the controllability results in several aspects: 1) we obtain a stronger sufficient condition for controllability of 2D NS system in an observed component and for L2- approximate controllability; 2) we prove that these criteria are valid for the case of ideal incompressible uid (\nu = 0); 3) we study solid controllability in projection on any finite-dimensional subspace and establish a sufficient criterion for such controllability

Necessary Optimality Conditions for a Dead Oil Isotherm Optimal Control Problem

We study a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of the mechanics of a continuous medium. Recent results on the problem provide existence, uniqueness and regularity of the optimal solution. Here we obtain the first necessary optimality conditions.Comment: 9 page

A simple proof of the approximate controllability from the interior for nonlinear evolution problems

The approximate controllability property for solutions of a large class of nonlinear evolution problems is obtained under some abstract conditions which hold, for instance, when the control is the right hand side of the equation. Our very simple method put in evidence the independence between the solvability of a boundary value problems and the study of the approximate controllability property which takes places in a number of cases. No duality type arguments are used which allows the consideration of very general nonlinear problems

Null controllability in a heat-solid structure model

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Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation

The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control. They assume that the internal control is only time dependent. The proof relies on the return method in combination with some local controllability results for nondegenerate equations and rescaling techniques