ABSTRACT
The linear-quadratic optimal control theory, introduced in 1960 by R.E. Kalman and R. S. Bucy [8], [9], has since that time enjoyed wide popularity as a mathematical framework in terms of which a wide variety of design objectives can be expressed and corresponding design techniques developed. This theory, further developed in an enormous number of journal articles and various books (see, e.g., [1]j[10],[19]) is so familiar that we need not give a systematic description of its contents in this paper. Rather, we will cite results from that theory as required to support our basic objective, the extension of that theory to nonlinear systems and to non-quadratic cost functionals in order to achieve improved performance and larger regions of asymptotic stability in the nonlinear system context. This work will be carried out primarily in neighborhoods of invariant sets of such systems, notably critical points and periodic solutions. In the process we will have reason to develop certain variations of the stable manifold theorem [4],[5] of nonlinear differential equations theory, including some new approaches to proving such theorems.
