| Details: |
Abstract:
Part I:
We will describe :
(i)What is Controllability problem
(ii) Examples and known results: ODE (finite dim),
transport equation, Heat equation (infinite dim ).
[This part of the talk will be
accessible to a broad section of student community and it will help to
understand the 2nd part]
Part II:
Then we consider compressible Navier-Stokes equations in one dimension,
linearized around a constant steady state $(Q_0,V_0)$, with
$Q_0 > 0,V_0\geq 0$. It is a coupled system involving both transport and
parabolic effects. We study the controllability of this linearized system
in bounded interval $(0,L)$. We find that the properties of the two
semigroups $(e^{tA})_{t\geq0} $ (the one when $V_0 = 0$ and the one when
$V_0> 0$) and the spectrum of $A$ are completely different where $A$ is
the corresponding linearized operator. We obtain several interesting
positive and negative results for the null controllability and approximate
controllability of the system using interior or boundary control in both
the cases $V_0 =0$ and $V_0>0$. |