The additive inverse eigenvalue problem over an algebraically closed field \(F\) of characteristic zero is considered: given a matrix \(A\in\text{gl}(n,F)\) and a matrix Lie subalgebra \({\mathcal L}\subset\text{gl}(n,F)\), under which conditions on \({\mathcal L}\) one can arbitrarily assign the eigenvalues of \(A+L\), when the perturbation \(L\) belongs to \({\mathcal L}\)? This problem is motivated by some important examples in control theory as the pole assignment or the dynamic compensation of a linear system. Using \textit{S. Friendland}'s theorem [Israel J. Math. 11, 184-189 (1972; Zbl 0252.15004); for the case \({\mathcal L}={\mathcal D}_ n\), the Lie algebra of diagonal matrices], it is shown that the additive inverse eigenvalue problem has a solution for any matrix \(A\) if and only if \(\text{rank} {\mathcal L}=n\) and some element of \({\mathcal L}\) has distinct eigenvalues.