  Navier-Stokes equations are well known in modelling of an incompressible
Newtonian fluid, such as air or water. This system of equations is very complex
due to the non-linearity term that characterizes it. After the linearization
and the discretization parts, we get a descriptor system of index-2 described
by a set of differential algebraic equations (DAEs). The two main parts we
develop through this paper are focused firstly on constructing an efficient
algorithm based on a projection technique onto an extended block Krylov
subspace, that appropriately allows us to construct a reduced system of the
original DAE system. Secondly, we solve a Linear Quadratic Regulator (LQR)
problem based on a Riccati feedback approach. This approach uses numerical
solutions of large-scale algebraic Riccati equations. To this end, we use the
extended Krylov subspace method that allows us to project the initial large
matrix problem onto a low order one that is solved by some direct methods.
These numerical solutions are used to obtain a feedback matrix that will be
used to stabilize the original system. We conclude by providing some numerical
results to confirm the performances of our proposed method compared to other
known methods.
