A.V. Fursikov

Boundary Value Problems and Optimal Boundary Control for the Navier--Stokes System: the Two-Dimensional Case

We study optimal boundary control problems for the two-dimensional Navier--Stokes equations in an unbounded domain. Control is effected through the Dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates. An objective of interest is the drag functional. We first establish three important results for inhomogeneous boundary value problems for the Navier--Stokes equations; namely, we identify the trace space for the velocity fields possessing finite energy, we prove the existence of a solution for the Navier--Stokes equations with boundary data belonging to the trace space, and we identify the space in which the stress vector (along the boundary) of admissible solutions is well defined. Then, we prove the existence of an optimal solution over the control set. Finally, we justify the use of Lagrange multiplier principles, derive an optimality system of equations in the weak sense from which optimal states and controls may be determined, and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values.

Exact boundary zero controllability of three-dimensional Navier-Stokes equations

abstractIn a bounded three-dimensional domain Ω a solenoidal initial vector fieldv0(x)∈H3 (Ω) is given. We construct a vector fieldz(t, x) defined on the lateral surface [0,T]×ϖΩ of the cylinder [0,T]×Ω which possesses the following property: the solutionv(t, x) of the boundary value problem for the Navier-Stokes equation with the initial valuev0(x) and the boundary Dirichlet conditionz(t, x) satisfies the relationv(T, x)≡0 at the instantT. Moreover,

On Controllability of Certain Systems Simulating a Fluid Flow

\parallel v(t, \cdot )\parallel _{H^3 (\Omega )} \leqslant c\exp \left( { - k/(T - t)^2 } \right) as t \to T

Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications

, wherec>0,k>0 are certain constants.

Instability in Models Connected with Fluid Flows II

We study the local exact boundary controllability problem for the Boussinesq equations that describe an incompressible fluid flow coupled to thermal dynamics. The result that we get in this paper is as follows: suppose that

On exact boundary zero-controlability of two-dimensional Navier-Stokes equations

\hat y(t,x)

Feedback stabilization for the Navier-Stokes equations: theory and calculations

is a given solution of the Boussinesq equation where

Approximate controllability of the Stokes system on cylinders by external unidirectional forces

t \in (0,T)