The hyperbolic map and applications to the linear quadratic regulator
In this textbook the hyperbolic map with some application is considered. By the hyperbolic map one means the mapping \(Y_{k+1}=[AY_ k+B][CY_ k+D]^{-1}\), where A,B,C,D and \(Y_ k\) are matrices. This book meanly discuss the scalar case, \(Y_ k=y_ k\). After the introduction, the authors give detailed analysis of the hyperbolic map both qualitatively and quantitatively. In chapter two the trajectory type of \(y_ k\) is predicted, and it is shown that although it can be very wild it is never chaotic. Using the simple idea of writing \(y_ k=u_ k/v_ k\) the hyperbolic map is rewritten to a linear system. With this linear system the exact solution of the hyperbolic map is derived. A general extension to the matrix case is given in the first appendix of chapter three. The second appendix of this chapter contains some interesting examples of problems in which the hyperbolic map occurs naturally. The well known example of the linear quadratic regular problem for discrete linear systems is treated in more detail in chapter four and five. In chapter four the relation between the regulator problem and the hyperbolic map is discussed. In the following chapter the analysis of the first chapters is used to discuss the behaviour of the hyperbolic map, which occurs in the regulator problem, and the state trajectories of the system for negative weighting of the state, i.e. Q is negative. Although the book contains no rigorous proofs, it is easy to read. Especially, since all results are illustrated, the book contains almost one hundred figures.
scientific article; zbMATH DE number 195216