In a recent paper () given one-parameter families of linear-quadratic control problems (with side condition that the state trajectory should vanish at infinity) have been investigated. It was proven there that, under two rather acceptable assumptions, the optimal cost depends continuously on the parameter. Moreover, optimal inputs (whenever they exist), state trajectories and outputs are continuous w.r.t. the parameter if the underlying systems are left invertible. However, in contrast with these problems with stability there generally proved to be no such continuity properties for problems without stability (i.e. the free end-point problems). In the present paper we will explain why. We will demonstrate that the definition of a new type of control problem with "partial" stability is necessary. The optimal cost for the "perturbed" problem then turns out to converge to the cost for this new problem. Additional results are found for inputs, states and outputs in case of left-invertibility. These results are established only by applying the assumptions made in the article mentioned above. Actually even less.