Minimum energy problems involving linear systems with quadratic performance criteria are classical in optimal control theory. The case where controls are constrained is discussed in Athans and Falb (1966) [Athans, M. and Falb, P.L. (1966), Optimal Control: An Introduction to the Theory and Its Applications, New York: McGraw-Hill Book Co.] who obtain a componentwise optimal control expression involving a saturation function expression. We show why the given expression is not generally optimal in the case where the dimension of the control is greater than one and provide a numerical counterexample.