Abstract The computation of approximate controls for the terminal problem for linear systems is considered. These controls can be made to steer the terminal state vector arbitrarily close to the desired terminal position; their norms will generally be smaller than the norm of the exact optimal control which achieves the desired terminal state. It is shown that such approximate controls can be obtained by means of constants b j ∗ such that an expression of the type ‖z− ∑ j=1 n−1 bjxj‖ takes on a value near to its minimum. By means of some arguments based upon the fact that problems of this kind are related to maximization problems in the corresponding dual space, it is proved that the constants b j ∗ satisfy an algebraic equation. The numerical computation of solutions of this equation is discussed briefly.