# Link between exact null controllability of two systems

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

H. O. Fattorini, Some remarks on complete controllability, Siam J. Control, 4 (1966), pp. 686-694.

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.

• Thank you. Could you please clarify the last claim, say for heat equation on $L^2(\Omega)$ e.g., why (2) is not exactly null controllable ? Feb 28, 2021 at 17:29
• If the initial condition has a singularity in the uncontrolled region, it will persist in time, because $R(\lambda_0,A)x$ is smoother than $x$ and $Bu$ is zero at the location of the singularity. Feb 28, 2021 at 19:28