On the classical solution to the linear-constrained minimum energy problem

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.