A new algorithm is presented for providing under-estimates of the reachable set from the origin for a class of n-dimensional linear systems with bounded controls. This algorithm is based on the novel approach of choosing a feedback control which makes all the eigenvalues of the closed loop system unstable. Results from feedback control and Liapunov stability theory are then used to formulate the problem as the minimization of a nonlinear function subject to constraints on certain matrices. The solution of this optimization problem provides an under-estimate of the reachable set in the form of an n-dimensional ellipsoid. Examples of both continuous and discrete-time systems are presented to illustrate the method. Comparison with existing exact results for some 2-dimensional systems shows that the method provides good approximations in these cases.