17h00 : Luz De Teresa (Univ. Mexico)
Some results on the controllability of coupled scalar parabolic equations.
In
this talk we will discuss some results on the null controllability of
coupled scalar parabolic equations. We will discuss several problems
related with the fact that the coupling matrix in the principal part of
the operator is not the identity. In one hand we will see the
difficulties arising in the boundary controllability of two one
dimensional equations when the coupling matrix is diagonal but not a
constant times the identity, and on the other hand, we will discuss
some results when trying to control with a distributed control and
acting, possibly, on each equation but when the coupling matrix in the
principal part is not diagonalizable. That is, in the second part of
the talk we deal with the controllability properties of some
nondiagonalizable parabolic systems.
Let Ω ⊂ R^N be a non-empty regular and bounded domain, let us fix T
> 0 and let us set Q:=Ω×(0,T) and Σ:=∂Ω×(0,T). We will consider the
null controllability properties of the linear system
y_t − A ∆ y = M(x,t) y + Bv1_ω in Q, y=0 on Σ, y(x,0)=y0(x) in Ω,
where ω ⊂ Ω is a (small) open subdomain, A∈L(R^n;R^n),
M∈L^∞(Q;L(R^n;R^n)), B∈L(R^m;R^n) and y0∈L^2(Ω)^n, with A a non
diagonalizable matrix.