The Riccati equation
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 [\textit{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 (\textit{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 (\textit{P. Lancaster} and \textit{L. Rodman}, Solutions of the continuous and discrete time algebraic Riccati equations: A review, pp. 11-51) and Chapter 3 (\textit{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 (\textit{A. Shayman}, A geometric view of the matrix Riccati equation, pp. 89-112) and Chapter 5 (\textit{C. Martin} and \textit{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 (\textit{S. Bittani}, \textit{P. Colerani} and \textit{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 (\textit{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 (\textit{H. L. Trentelman} and \textit{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 (\textit{J. L. Willems} and \textit{F. M. Callier}, The infinite horizon and the receding horizon LQ-problems with partial stabilization constraints, pp. 243-262) and Chapter 10 (\textit{R. R. Bitmead} and \textit{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 (\textit{T. Başar}, General Riccati equation in dynamic games, pp. 293-333).
stability theory for feedback control systems
scientific article; zbMATH DE number 51782