In this thesis, we analyse linear systems of ordinary differential equations, which are parametrised by \(\nu \in K\), where \(K\)is a compact set (in our case, \(K \subset \mathbb{R}^{n}\), for some \(n\)). In applications, we want to construct a function \(u_{\nu}\), called an input function, which for a given \(\nu \in K\), drives the state function \(x(t)\) to state \(x^{1}\) at \(t=T\), where \(x^{1}\) is determined beforehand. Such a construction is described in the proof of Theorem 1.5.3. Numerically, to construct such a function, we need to solve a linear system of equations (2.11), which can take a long time, and the matrix can be ill-conditioned. So, we wish to determine a finite number of representative input functions \(u_{1},...,u_{n}\) (which span a space \(V_{n} \subset (L^{2}[0,T])^{m}\)) such that for every \(\nu \in K\), we can approximate the relevant input function in the space \(V_{n}\) in the way that the following holds: \[||x(T)-x^{1}||<\epsilon,\] where \(\epsilon\) is fixed. So, we wish to speed up the process of find the input function, but we lose the precision by requiring approximate controllability instead of exact controllability. In the last chapter, the proposed method is shown on two examples: for the wave equation and for the heat equation.