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Second-order conditions for steady-state optimality and nonoptimality in a periodic control problem are presented. The main result is a generalization of the π test, a second-order sufficient condition for improved performance by periodic control. Earlier results are generalized in two distinct ways: 1) the control constraint set is only assumed to be convex (and, hence, possibly nonopen) thus allowing the optimal steady-slate control to be an element of the control constraint set boundary; and 2) auxiliary normality conditions are eliminated. Proofs of the results are based upon second-order necessary, conditions for nonlinear programming and optimal control obtained recently in  and .