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Book review

Author(s) Bensoussan, Alain
Title Stochastic Control of Partially observable Systems
Publisher Cambridge University Press
Year of publication 1992
Reviewed by Ilie Parpucea

In many applications, the problem of stochastic control of partially observable systems plays an important role. This justifies the importance of having a theory as complete as possible, which can be used for numerical implementation.

This book first presents those problems which may be dealt with algebraically under the linear theory. Later chapters discuss nonlinear filtering theory, in which the statistics are infinite dimensional, and from this approximations and perturbation methods are developed.

In the first three chapters of this book are presented problems which can be dealt with directly by algebraic manipulations, without using the complete theory.

In Chapters 4 to 6, are presented the theory of non linear filtering, which is basic step in formulating the control problem adequately. The main difficulty, especially from the point of view of numerical applications, is that there are no statistics which are finite dimensional, and the basic object to be computed is the conditional probability.

Chapter 7 is intended to study stochastic control problems with partial information, in an intermediate case, namely when the direct methods of chapters 1, 2, 3 are not applicable yet the full theory is not necessary, either because finite dimensional sufficient statistics are available, or approximations are possible. In Chapter 8 are presented the stochastic maximum principle and dynamic programming approach to the problem of stochastic control with partial information in the general case, which implies infinite dimensionality.

In Chapter 9 are stated, in a limited framework, some existence results.

Reference to relevant literature is given throughout the book, for further study of these aspects.

If your personal and/or institutional library is still a bit thin in the important area of stochastic control, this is a volume well worth investing in.

A. Bensoussan is the leading world authority on the subject and this volume will be a valuable reference to all those working in stochastic oontrol and filtering theory.