##
**Rational matrix equations in stochastic control.**
*(English)*
Zbl 1034.93001

Lecture Notes in Control and Information Sciences 297. Berlin: Springer (ISBN 3-540-20516-0/pbk). xv, 200 p. (2004).

This monograph arises from the dissertation of the author.

Some results on the existence of solutions to Riccati and Lyapunov equations as well as on convergence to these solutions using a Newton iteration algorithm are obtained.

An important portion of the work presents the basic ingredients: First the author recalls stabilization ideas in a stochastic context. The Riccati equation which appears in the first chapter is explained next when presenting optimal stabilization and disturbance rejection. Chapter three reviews notions on ordered function spaces. The notion of resolvent positive operators will be of importance. In such an ordered function space and associated cone \(C\), a bounded operator \(T\) is resolvent positive if \((\alpha\text{Id}-T)^{-1}\) is bounded and leaves \(C\) invariant for all \(\alpha\) reals greater than an certain \(\alpha_0\). It is seen that the Lyapunov operator is resolvent position, and the (mean-square) stability of a linear stochastic system is equivalent to resolvent positivity of an associated Lyapunov-type operator. In the next chapter, Newton’s method in an operator theoretic context is recalled, and nonlocal convergence results of Riccati operators using the notions of concavity and resolvent positivity are established. In chapter five, the solution to the Riccati equation with indefinite constraints is obtained via a duality method which allows one to recover the concavity assumption of before.

There are 221 references, a modest index and a list of notations at the end.

The contributions of this work, when suitably compressed, would constitute excellent publications. In this book format, the presentation does not permit these contributions to be put in the context of the thousands of published works on the Riccati equation. The context of the author (the reminders) serve the interest of the cause and support the establishment of the specific results.

Some results on the existence of solutions to Riccati and Lyapunov equations as well as on convergence to these solutions using a Newton iteration algorithm are obtained.

An important portion of the work presents the basic ingredients: First the author recalls stabilization ideas in a stochastic context. The Riccati equation which appears in the first chapter is explained next when presenting optimal stabilization and disturbance rejection. Chapter three reviews notions on ordered function spaces. The notion of resolvent positive operators will be of importance. In such an ordered function space and associated cone \(C\), a bounded operator \(T\) is resolvent positive if \((\alpha\text{Id}-T)^{-1}\) is bounded and leaves \(C\) invariant for all \(\alpha\) reals greater than an certain \(\alpha_0\). It is seen that the Lyapunov operator is resolvent position, and the (mean-square) stability of a linear stochastic system is equivalent to resolvent positivity of an associated Lyapunov-type operator. In the next chapter, Newton’s method in an operator theoretic context is recalled, and nonlocal convergence results of Riccati operators using the notions of concavity and resolvent positivity are established. In chapter five, the solution to the Riccati equation with indefinite constraints is obtained via a duality method which allows one to recover the concavity assumption of before.

There are 221 references, a modest index and a list of notations at the end.

The contributions of this work, when suitably compressed, would constitute excellent publications. In this book format, the presentation does not permit these contributions to be put in the context of the thousands of published works on the Riccati equation. The context of the author (the reminders) serve the interest of the cause and support the establishment of the specific results.

Reviewer: A. Akutowicz (Berlin)

### MSC:

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

93E15 | Stochastic stability in control theory |

93D15 | Stabilization of systems by feedback |

93B40 | Computational methods in systems theory (MSC2010) |

93B28 | Operator-theoretic methods |

47J25 | Iterative procedures involving nonlinear operators |

93E20 | Optimal stochastic control |

49N10 | Linear-quadratic optimal control problems |