Let $A: D(A) \subset H \rightarrow H$ generate a strongly continuous semigroup $T(t)$ on a Hilbert space $H$ and $B\in \mathcal{B}(H)$. Consider the two control systems:
$$(1)\; x'(t)=Ax(t)+ Bu(t) \qquad \text{ and } \qquad (2)\; x'(t)=R(\lambda\_0,A)x(t)+Bu(t),$$
where $\lambda\_0 \in \rho(A): \Re \lambda\_0 \ge\omega>\omega\_0(T)$ (the type of $T(t)$).


In



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> H. O. Fattorini, Some remarks on complete controllability, Siam J. Control, 4 (1966), pp. 686-694.
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it is proved that the approximate controllability of $(1)$ and of $(2)$ are equivalent. See for instance Proposition 2.3.


My question: is there any relation between exact null controllability of $(1)$ and of $(2)$? If so, any reference that consider this topic would be helpful.



Link between exact null controllability of two systems
No. Many PDE systems, e.g. the heat and wave equation, allow exact null controllability for controls restricted to a proper subset of the physical domain. On the other hand, resolvent operators are smoothing, so if the initial condition for (2) has a singularity outside the controlled region, there is no way for the control to remove it.


