##
**Quadratic functionals in variational analysis and control theory.**
*(English)*
Zbl 0842.49001

Mathematical Topics. 6. Berlin: Akademie-Verlag. 293 p. (1995).

The major focus of this book is on continuous-time linear Hamiltonian (selfadjoint) differential systems and associated linear quadratic optimal control (variational) problems. The differential systems and quadratic functionals are only considered on a finite time-interval. This book contains many recent results which extend and generalize the existing theory in the field. The level of presentation is mathematically rigorous but self-contained in the sense that it is accessible to graduate students in mathematics.

In Chapter 0 a preview is given by means of a very simple example of the problems treated in the book, namely Sturmian eigenvalue problems. Chapter 1 gives a preliminary treatment of Hamiltonian systems and Picone’s identity in relation with variational calculus. Homogeneous and inhomogeneous quadratic functionals are considered in Chapter 2. The intent of Chapter 3 is to provide recent results from matrix theory, essentially from the author himself, which are crucial in deriving the results in the rest of the book. In Chapter 4 Hamiltonian systems are considered in depth. An important notion is oscillation, which has here to do with the number of time instants for which a certain matrix-valued function of time is not invertible. Riccati matrix differential equations are studied in Chapters 5 and 6, where inequalities, index results and asymptotics are the subjects. Then in Chapter 7 the main results in the book are stated. The oscillation results above are extended, Sturmian theory is generalized and comparison theorems are stated, to name a few. Finally, Chapter 8 is devoted to applications of the theory in the former chapters. One of the applications is the optimal linear regulator.

The present book is clearly written and well organized. Apart from the above-mentioned self-containedness, basic definitions/assumptions/formulae are repeated in different chapters for the convenience of the reader. One of the consequences is that the main Chapters 7 and 8 can be understood without having read the other parts of the book. A very helpful commentary concludes each chapter and provides an extensive guide to the literature. Many results presented were recently developed by the author and, together with other material, make up a modern account of some of the latest research in this area. All this makes the book to an excellent advanced graduate text for students in mathematics and as a research monograph rich in ideas for mathematically oriented specialists in areas such as variational analysis and control theory. For the practicing (control) engineer this book is too academic, I’m afraid.

In Chapter 0 a preview is given by means of a very simple example of the problems treated in the book, namely Sturmian eigenvalue problems. Chapter 1 gives a preliminary treatment of Hamiltonian systems and Picone’s identity in relation with variational calculus. Homogeneous and inhomogeneous quadratic functionals are considered in Chapter 2. The intent of Chapter 3 is to provide recent results from matrix theory, essentially from the author himself, which are crucial in deriving the results in the rest of the book. In Chapter 4 Hamiltonian systems are considered in depth. An important notion is oscillation, which has here to do with the number of time instants for which a certain matrix-valued function of time is not invertible. Riccati matrix differential equations are studied in Chapters 5 and 6, where inequalities, index results and asymptotics are the subjects. Then in Chapter 7 the main results in the book are stated. The oscillation results above are extended, Sturmian theory is generalized and comparison theorems are stated, to name a few. Finally, Chapter 8 is devoted to applications of the theory in the former chapters. One of the applications is the optimal linear regulator.

The present book is clearly written and well organized. Apart from the above-mentioned self-containedness, basic definitions/assumptions/formulae are repeated in different chapters for the convenience of the reader. One of the consequences is that the main Chapters 7 and 8 can be understood without having read the other parts of the book. A very helpful commentary concludes each chapter and provides an extensive guide to the literature. Many results presented were recently developed by the author and, together with other material, make up a modern account of some of the latest research in this area. All this makes the book to an excellent advanced graduate text for students in mathematics and as a research monograph rich in ideas for mathematically oriented specialists in areas such as variational analysis and control theory. For the practicing (control) engineer this book is too academic, I’m afraid.

Reviewer: W.L.de Koning (Delft)

### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

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

34B24 | Sturm-Liouville theory |

34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |

49N10 | Linear-quadratic optimal control problems |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |