##
**Differential equations, discrete systems and control. Economic models.**
*(English)*
Zbl 0912.34002

Mathematical Modelling: Theory and Applications. 3. Dordrecht: Kluwer Academic Publishers. xvi, 357 p. (1997).

This book represents an interesting, clear and complete exposition of methods of dynamical systems and control theory. The leitmotiv is provided by applications to economic models, based on ordinary differential and difference equations. A remarkable feature of the book is that for every theoretical result, both the continuous time and the discrete time version is presented. Continuous and discrete versions are given for most of the economic models considered in this book.

The reader is not assumed to be familiar with ordinary differential equations. The exposition is organized in such a way to introduce the simplest types of differential equations and their solution techniques at the beginning, and to proceed therefore to differential equations of an increasing degree of difficulty. Each type of equation is motivated by appropriate examples. The economic meaning of solutions is always discussed.

In particular, Chapter 1 to 3 deal with linear equations and systems of linear equations. Chapter 4 contains some results about existence and uniqueness of general nonlinear systems. Moreover, some important qualitative notions (equilibria, stability) are introduced and analyzed. Because of their interest for computer implementation, some numerical methods for differential equations are illustrated in Chapter 5. Chapter 6, 7, and 8 are devoted to stabilization and the linear quadratic optimization problem: the authors consider both the finite and the infinite horizon case. Finally, Chapter 9 and 10 deal with some general optimal control problems. Several economic models are studied by means of the classical Euler-Lagrange equations and the Pontryagin maximum principle.

Although the book is basically written for tutorial purposes, it can be a nice and useful reading for researchers both in the field of mathematics and economics.

The reader is not assumed to be familiar with ordinary differential equations. The exposition is organized in such a way to introduce the simplest types of differential equations and their solution techniques at the beginning, and to proceed therefore to differential equations of an increasing degree of difficulty. Each type of equation is motivated by appropriate examples. The economic meaning of solutions is always discussed.

In particular, Chapter 1 to 3 deal with linear equations and systems of linear equations. Chapter 4 contains some results about existence and uniqueness of general nonlinear systems. Moreover, some important qualitative notions (equilibria, stability) are introduced and analyzed. Because of their interest for computer implementation, some numerical methods for differential equations are illustrated in Chapter 5. Chapter 6, 7, and 8 are devoted to stabilization and the linear quadratic optimization problem: the authors consider both the finite and the infinite horizon case. Finally, Chapter 9 and 10 deal with some general optimal control problems. Several economic models are studied by means of the classical Euler-Lagrange equations and the Pontryagin maximum principle.

Although the book is basically written for tutorial purposes, it can be a nice and useful reading for researchers both in the field of mathematics and economics.

Reviewer: A.Bacciotti (Torino)

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

34H05 | Control problems involving ordinary differential equations |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |