In optimization problems, it is urged from practical points of view to develop effective synthesis methods based on an approximate model obtainable by reducing the system dimension. For instance, the numerical calculation of optimal control needs enormous endeavor with increase of dimension of the control system, because the generated canonical system equation is “2~-dimensional” when the original system is “n-dimensional”. Moreover, the essential difficulty lies in that one must solve it under two-point boundary conditions. Various methods have been discussed on solving the boundary value problem, eg, sweep method [I], shooting method [2], quasilinearization [3], invariant imbedding [4], etc. This paper gives a powerful tool for the analysis of the fixed-terminal minimum energy problems with the aid of the Singular Perturbation Theory [5-71 and the Riccati transformation [8]. The Riccati transformation is efficient in computation of the so-called ill-conditioned two-point boundary value problem. The Singular Perturbation Theory has two aspects: one is to consider the convergence of the solution of the full system to that of the reduced system as a small parameter, whose existence makes the system order higher, tends to zero; the other is to construct an asymptotic expansion of the solution, which can offer a desired approximate solution by truncation. In both the aspects, the stability of the boundary layer system plays a crucial role [5-71.

Kokotovic et al. first used the Singular Perturbation Theory in developing a method to reduce the system order in optimization problems [9]. In their studies of the linear regulator problem, it is needed that the boundary layer system should be asymptotically stable [9], or that the state matrix of the boundary layer system should be stable and the full system should satisfy a peculiar condition [IO], or that the boundary layer system should be controllable and observable [111.