An asymptotic expansion for an optimal relaxation oscillator

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 expansion is divided into two parts, an outer part at the time scale of the period and an inner part characterized by an even faster time scale which captures the fast transitions. These two solutions are matched together to obtain the resulting asymptotic solution in which the performance index and the optimal period are obtained up to third order, and the states and the control are obtained up to second order.