For certain reduced-order optimization problems where assumed fast dynamics are neglected, chattering optimal solutions occur. The chattering optimal solution is represented by some of the variables alternating between distinctively different :values at an infinite rate. For a simple and somewhat transparent periodic optimal control problem, the neglected dynamics are included by an asymptotic expansion about the chattering solution. The periodic chattering arc is approached as a weighting parameter, associated with the control penalty in the performance index, goes toward zero. This weighting parameter is used, as the expansion parameter to form an asymptotic expansion about the chattering arc. In particular, two time scales are used in the expansions. A time scale proportional to the period is used to transform the problem to one similar to that of a relaxation oscillator where the problem is characterized by slow, almost equilibrium motions connected by fast, jump type transitions. The asymptotic expan...