Cheap and singular controls for linear quadratic regulators

A linear system with quadratic cost function where a small parameter ??2 multiplies the control cost is considered. Due to the cheapness of control, a strong control action in the form of high-gain feedback forces the given system to have slow and fast transients of a hierarchy of time-scales coupled with high and low amplitude interaction. By appropriate amplitude and time-scaling of variables, these interactions are normalized and the considered problem is decomposed into several nonsingular subproblems of minimal order, each pertaining to only one time-scale. Complete results characterizing the limiting behavior of optimal performance index, eigenvalues, trajectory and control variables as ?? ?? 0 are given. All the different ways by which nonuniqueness can pop into singular control are discussed. More importantly, the method developed here allows the design of a high-gain feedback system in terms of the design of several lower order well defined subproblems. Thus it gives a strong impeteus for practical implementation of the theory developed.