The Riccati equation.

*(English)*Zbl 0734.34004
Communications and Control Engineering Series. Berlin etc.: Springer- Verlag. x, 338 p. DM 158.00/hbk (1991).

From June 26-28, 1989, a ‘Workshop on the Riccati equation in control, systems, and signals’ was held in Como, Italy. The proceedings of the workshop have already appeared elsewhere [S. Bittani (ed.), Proc. Workshop on the Riccati equation in control, systems, and signals, Como, Italy. Pitagora Editrice Bologna (1989)], but independently the workshop “constituted the foundation from which the idea of this book germinated. It presents a self-contained treatment of the main issues evolving around the Riccati equation, in particular theory, applications, and numerical algorithms”. The book consists of 11 coordinated tutorial chapters written by different authors; it is intended as a graduate text as well as a reference for scientists, especially engineers, and mathematicians. The organization is as follows:

Chapter 1 (S. Bittani, Count Riccati and the early days of the Riccati equation, pp. 1-10) is devoted to the history and pre-history of the Riccati equation. Chapter 2 (P. Lancaster and L. Rodman, Solutions of the continuous and discrete time algebraic Riccati equations: A review, pp. 11-51) and Chapter 3 (V. Kučera, Algebraic Riccati equations: Hermitian and definite solutions, pp. 53-88) supply a comprehensive view of the algebraic Riccati equation, mainly based on a linear algebra approach. A geometrical analysis of the equation is carried out in Chapter 4 (A. Shayman, A geometric view of the matrix Riccati equation, pp. 89-112) and Chapter 5 (C. Martin and G. Ammar, The geometry of the matrix Riccati equation and associated eigenvalue methods, pp. 113-126). Chapters 2 to 5 deal with the constant coefficient case.

The periodically time-varying Riccati equation is the subject of Chapter 6 (S. Bittani, P. Colerani and G. de Nicolao, The periodic Riccati equation, pp. 127-162). The leading numerical techniques for the solution of the Riccati equation are overviewed in Chapter 7 (A. J. Laub, Invariant subspace methods for the numerical solution of Riccati equations, pp. 163-196, presenting 237 references).

The remaining four chapters address connections between the Riccati equation and some important problems in systems and control. More precisely, in Chapter 8 (H. L. Trentelman and J. C. Willems, The dissipation inequality and the algebraic Riccati equation, pp. 197- 242), the role of the Riccati equation in the study of dissipative systems is elucidated, including applications to linear-quadratic optimal control problems, stability theory for feedback control systems, electrical network synthesis, covariance generation for stationary finite-dimensional Gauss-Markov processes, and to the \(H_{\infty}\) control problem. The linear-quadratic optimal control problem in its various facets is the subject of Chapter 9 (J. L. Willems and F. M. Callier, The infinite horizon and the receding horizon LQ-problems with partial stabilization constraints, pp. 243-262) and Chapter 10 (R. R. Bitmead and M. Gevers, Riccati difference and differential equations: Convergence, monotonicity and stability, pp. 263-291). Finally, a unified survey on generalized Riccati equations in dynamic games is presented in Chapter 11 (T. Başar, General Riccati equation in dynamic games, pp. 293-333).

Chapter 1 (S. Bittani, Count Riccati and the early days of the Riccati equation, pp. 1-10) is devoted to the history and pre-history of the Riccati equation. Chapter 2 (P. Lancaster and L. Rodman, Solutions of the continuous and discrete time algebraic Riccati equations: A review, pp. 11-51) and Chapter 3 (V. Kučera, Algebraic Riccati equations: Hermitian and definite solutions, pp. 53-88) supply a comprehensive view of the algebraic Riccati equation, mainly based on a linear algebra approach. A geometrical analysis of the equation is carried out in Chapter 4 (A. Shayman, A geometric view of the matrix Riccati equation, pp. 89-112) and Chapter 5 (C. Martin and G. Ammar, The geometry of the matrix Riccati equation and associated eigenvalue methods, pp. 113-126). Chapters 2 to 5 deal with the constant coefficient case.

The periodically time-varying Riccati equation is the subject of Chapter 6 (S. Bittani, P. Colerani and G. de Nicolao, The periodic Riccati equation, pp. 127-162). The leading numerical techniques for the solution of the Riccati equation are overviewed in Chapter 7 (A. J. Laub, Invariant subspace methods for the numerical solution of Riccati equations, pp. 163-196, presenting 237 references).

The remaining four chapters address connections between the Riccati equation and some important problems in systems and control. More precisely, in Chapter 8 (H. L. Trentelman and J. C. Willems, The dissipation inequality and the algebraic Riccati equation, pp. 197- 242), the role of the Riccati equation in the study of dissipative systems is elucidated, including applications to linear-quadratic optimal control problems, stability theory for feedback control systems, electrical network synthesis, covariance generation for stationary finite-dimensional Gauss-Markov processes, and to the \(H_{\infty}\) control problem. The linear-quadratic optimal control problem in its various facets is the subject of Chapter 9 (J. L. Willems and F. M. Callier, The infinite horizon and the receding horizon LQ-problems with partial stabilization constraints, pp. 243-262) and Chapter 10 (R. R. Bitmead and M. Gevers, Riccati difference and differential equations: Convergence, monotonicity and stability, pp. 263-291). Finally, a unified survey on generalized Riccati equations in dynamic games is presented in Chapter 11 (T. Başar, General Riccati equation in dynamic games, pp. 293-333).

Reviewer: W.Müller (Berlin)

##### MSC:

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

34A05 | Explicit solutions, first integrals of ordinary differential equations |

34A26 | Geometric methods in ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

15A24 | Matrix equations and identities |

65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |

65F30 | Other matrix algorithms (MSC2010) |

91A23 | Differential games (aspects of game theory) |

00B15 | Collections of articles of miscellaneous specific interest |

01A50 | History of mathematics in the 18th century |

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

93B25 | Algebraic methods |

93B36 | \(H^\infty\)-control |

93C05 | Linear systems in control theory |

93C55 | Discrete-time control/observation systems |

93D10 | Popov-type stability of feedback systems |

93D15 | Stabilization of systems by feedback |

93E11 | Filtering in stochastic control theory |