Suboptimal Control of Singularly Perturbed Systems and Periodic Optimization

A traditional approach to singularly perturbed optimal control problems is based on an approximation of these problems by some reduced problems which are obtained via the formal replacement of the fast variables by the states of equilibrium of the “fast subsystems” considered with “frozen” slow variables and controls. We show that such approximation may be not valid and that it is valid if and only if certain families of periodic optimization problems admit steady state solutions. We show also how the solutions of these problems can be used to construct suboptimal controls for singularly perturbed problems when the approximation by the reduced problems is not possible.