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This paper considers the penalty function method to obtain an approximate solution to the bounded phase coordinate optimal control problem for linear discrete systems with essentially quadratic cost functionals. The penalty function assumes positive values outside the phase constraint set, and zero inside the phase constraint set. The problem is to find an optimal control from a convex compact control restraint set such that the cost functional is minimum, and the sum of the penalty function along the response is smaller than a prescribed constant. It is shown that the maximum principle is a necessary and sufficient condition for an optimal control in a number of cases, and an analytic method of finding an optimal control is given. Also, the existence of an optimal control is proved.