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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).
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