We consider the finite-dimensional system \begin{equation}\begin{cases}y'(t)=Ay(t)+Bv,\ t\in (0,T)\\ y(0)=y^0\end{cases}\end{equation} Where $v$ is the control, $A\in Mat(N\times N); B\in Mat(N\times N)$ and $y^0\in \mathbb{R}^N$.
This problem is exact controllable if for any fixed $y^1\in\mathbb{R}^N; y^0\in\mathbb{R}^N$, there exists a control $v\in \mathbb{R}^N$ such that $y(T)=y^1$.
There are many ways to get the exact controllability of this problem. For examples: Kalman rank condition; (HUM): Observability inequality, Uniqueness continuation,....
We have read a paper which the author gave the following way:
"If $y^0=0$ then the finite-dimensional problem is exactly controllable if for $f\in \mathbb{R}^N$ and any $v\in \mathbb{R}^N$ such that $<y(T),f>=0$ then $f\equiv 0$."
So, in the case $y^0\neq 0$, What is the similar?