Matrix Riccati equations in control and systems theory.

*(English)*Zbl 1027.93001The book treats differential Riccati equations
\[
\dot W= M_{21}(t)+ M_{22}(t) W- WM_{11}(t)- WM_{12}(t) W,\qquad -\infty< t<\infty,
\]
difference Riccati equations
\[
W(k+1)=- M_{21}- M_{22}W(k)(I- M_{21}W(k))^{-1} M_{11},\qquad k= 0,\pm 1,\pm 2,\dots,
\]
and the corresponding algebraic Riccati equations
\[
0= M_{21}+ M_{22} W- WM_{11}- WM_{12} W
\]
and
\[
W= -M_{21}- M_{22} W(I- M_{21} W)^{-1} M_{11}.
\]
Here \(M_{ij}\) are given matrices (variable or constant, as the case may be) of suitable sizes. Most of the book is devoted to the differential Riccati equations and their algebraic counterparts.

The interest in Riccati equations has become huge, especially in recent decades, and the literature on the subject comprises thousands of items, including dozens of books. Much of the research in Riccati equations is motivated by applications in mathematics, science, and engineering. Several areas of applications, including robust control, disturbance attenuation, game theory, and stochastic control, are emphasized in the reviewed book. From the standpoint of theoretical mathematics, Riccati equations represent a relatively well understood, albeit not without its own intricacies and subtleties, class of nonlinear equations: They can be linearized via the Grassmannian formulation, and exhibit several remarkable properties (for example, preservation of the Loewner partial order on Hermitian matrices). A key issue regarding Riccati equations involves conditions that guarantee existence of global solutions, i.e., solutions that do not escape in finite time.

The reviewed book addresses basic properties of Riccati equations, several of their applications, and brings the exposition to the state of the art of very recent research. The subject matter is restricted to matrix equations (thus, leaving out Riccati equations with possibly unbounded operator coefficients, which are of primary importance in optimal control of systems governed by PDEs), and to mainly theoretical issues, thus de-emphasizing numerical methods (although several numerical algorithms and examples are presented). Within these restrictions, however, the book is a useful, timely, and welcome addition to the literature, as it brings for the first time many recent developments in a book form, especially on coupled and generalized equations. The book will definitely become a standard reference for many aspects of the more advanced theory and applications of matrix Riccati equations.

Chapter headings, with brief comments in parentheses, are as follows: 1. Basic results for linear equations. 2. Hamiltonian matrices and algebraic Riccati equations. (The first two chapters contain rather standard material which can be found in some form in many sources nowadays.) 3. Global aspects of Riccati differential and difference equations. (This chapter includes the study of flows on Grassmann manifolds and Riccati equations with periodic coefficients.) 4. Hermitian-Riccati differential equations. (In particular, a complete proof is presented of the theorem stating that the Riccati differential equation is essentially the only equation having the Loewner order preserving property.) 5. The periodic Riccati equation. (A description of positive semidefinite periodic equilibria is included.) 6. Coupled and generalized Riccati equations. (This chapter also contains applications to Nash and Stackelberg games, stochastic control, and Markovian jump systems. In Stackelberg games the players obey a hierarchical structure of leaders and followers, in contrast to Nash games.) 7. Symmetric differential Riccati equations: an operator based approach. (Here, the “Riccati theory”, i.e., to quote the book, “connections between stabilizing solutions to the symmetric algebraic Riccati equation, invertibility of the Toeplitz operator associated with the input-output operator of the underlying Hamiltonian system, and existence of antianalytic factorization of the Popov function”, is presented in the context of time-varying systems and the descriptor operator approach.) 8. Applications to robust control systems. (The material here contains a thorough treatment of the four-block Nehari and disturbance attenuation problems.) 9. Nonsymmetric Riccati theory and applications. (Extension of the Riccati theory to the nonsymmetric algebraic Riccati equation is developed, with applications in game theory, in particular, open loop Nash and Stackelberg equilibria.)

The up-to-date bibliography is quite extensive, containing more than 400 items.

The interest in Riccati equations has become huge, especially in recent decades, and the literature on the subject comprises thousands of items, including dozens of books. Much of the research in Riccati equations is motivated by applications in mathematics, science, and engineering. Several areas of applications, including robust control, disturbance attenuation, game theory, and stochastic control, are emphasized in the reviewed book. From the standpoint of theoretical mathematics, Riccati equations represent a relatively well understood, albeit not without its own intricacies and subtleties, class of nonlinear equations: They can be linearized via the Grassmannian formulation, and exhibit several remarkable properties (for example, preservation of the Loewner partial order on Hermitian matrices). A key issue regarding Riccati equations involves conditions that guarantee existence of global solutions, i.e., solutions that do not escape in finite time.

The reviewed book addresses basic properties of Riccati equations, several of their applications, and brings the exposition to the state of the art of very recent research. The subject matter is restricted to matrix equations (thus, leaving out Riccati equations with possibly unbounded operator coefficients, which are of primary importance in optimal control of systems governed by PDEs), and to mainly theoretical issues, thus de-emphasizing numerical methods (although several numerical algorithms and examples are presented). Within these restrictions, however, the book is a useful, timely, and welcome addition to the literature, as it brings for the first time many recent developments in a book form, especially on coupled and generalized equations. The book will definitely become a standard reference for many aspects of the more advanced theory and applications of matrix Riccati equations.

Chapter headings, with brief comments in parentheses, are as follows: 1. Basic results for linear equations. 2. Hamiltonian matrices and algebraic Riccati equations. (The first two chapters contain rather standard material which can be found in some form in many sources nowadays.) 3. Global aspects of Riccati differential and difference equations. (This chapter includes the study of flows on Grassmann manifolds and Riccati equations with periodic coefficients.) 4. Hermitian-Riccati differential equations. (In particular, a complete proof is presented of the theorem stating that the Riccati differential equation is essentially the only equation having the Loewner order preserving property.) 5. The periodic Riccati equation. (A description of positive semidefinite periodic equilibria is included.) 6. Coupled and generalized Riccati equations. (This chapter also contains applications to Nash and Stackelberg games, stochastic control, and Markovian jump systems. In Stackelberg games the players obey a hierarchical structure of leaders and followers, in contrast to Nash games.) 7. Symmetric differential Riccati equations: an operator based approach. (Here, the “Riccati theory”, i.e., to quote the book, “connections between stabilizing solutions to the symmetric algebraic Riccati equation, invertibility of the Toeplitz operator associated with the input-output operator of the underlying Hamiltonian system, and existence of antianalytic factorization of the Popov function”, is presented in the context of time-varying systems and the descriptor operator approach.) 8. Applications to robust control systems. (The material here contains a thorough treatment of the four-block Nehari and disturbance attenuation problems.) 9. Nonsymmetric Riccati theory and applications. (Extension of the Riccati theory to the nonsymmetric algebraic Riccati equation is developed, with applications in game theory, in particular, open loop Nash and Stackelberg equilibria.)

The up-to-date bibliography is quite extensive, containing more than 400 items.

Reviewer: L.Rodman (Williamsburg)

##### MSC:

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

93D15 | Stabilization of systems by feedback |

15A24 | Matrix equations and identities |

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

93B28 | Operator-theoretic methods |