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. The reduced problems 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. It is shown that such as approximation may not be valid, and that it is valid if and only if certain families of periodic optimization problems admit steady-state solutions. It is also shown 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