Multiple time-scale decomposition in cheap control problems--Singular control

A linear quadratic regulator problem where a small parameter μ
²
multiplies the control cost is considered. As
\mu \rightarrow 0
, it exhibits a dynamic behavior with multiple time-scales, whereas for
\mu = 0
, it is a singular control problem. At first a multiparameter singularly perturbed model of the given problem is constructed by transferring the singularity that exists in the performance index to the dynamic equations. Then the problem is decomposed into several subproblems of minimal order, each pertaining to only one time-scale. The method developed permits under a single framework the characterization of asymptotic behavior of optimal closed-loop poles, state and control trajectories, performance index, and optimal transfer function as
\mu \rightarrow 0
. Moreover, the asymptotic solution brings into focus 1) the nature of singular control and its connection to transmission zeros and 2) the presence of various types of impulses and other higher order distributions in both state and control. The approach unifies and extends significantly several ideas previously somewhat disjoint.