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**Viability theory.**
*(English)*
Zbl 0755.93003

Systems & Control: Foundations & Applications. Boston, MA etc.: Birkhäuser. xxv, 543 p., 14 ill. (1991).

This impressive book gives an account of the theory that has developed over the past few years, mainly under the guidance of the present author. Viability theory may be placed somewhere between the theory of differential inclusions, control theory and set valued analysis. The book by the author and H. Frankowska [Set valued analysis (1990; Zbl 0713.49021)] is taken in the present book as a standard reference (though its knowledge is not presupposed). In the author’s words, “the main purpose of viability theory is to explain evolution of a system, determined by nondeterministic dynamics and viability constraints, to reveal the concealed feedbacks which allow the system to be regulated, and provide selection mechanisms for implementing them.”

The book starts with a chapter recasting parts of the classical theory of ordinary stochastic (white noise driven) systems from the viewpoint of viability theory. This serves mainly motivational purposes. The second chapter gives an exposition of fundamentals on set-valued maps. Chapters 3-7 present the core of viability theory for differential inclusions: The basic viability theorems, viability kernels and exit tubes, invariance theorems, regulation of control systems, smooth and heavy solutions. Chapter 8 discusses the tracking problem and related problems from control theory. Lyapunov functions are dealt with in Chapter 0, various issues along them finite difference schemes are discussed in Chapter 10. Chapter 11 introduces viability tubes describing viability constraints changing with time. The final Chapters 12-14 deal with extensions to functional differential inclusions, partial differential inclusions and differential games.

The author takes great care in motivating and explaining the relevance of viability theory in general and of the contents of this book. In my opinion, mathematics will greatly benefit from more books written in this style.

The book starts with a chapter recasting parts of the classical theory of ordinary stochastic (white noise driven) systems from the viewpoint of viability theory. This serves mainly motivational purposes. The second chapter gives an exposition of fundamentals on set-valued maps. Chapters 3-7 present the core of viability theory for differential inclusions: The basic viability theorems, viability kernels and exit tubes, invariance theorems, regulation of control systems, smooth and heavy solutions. Chapter 8 discusses the tracking problem and related problems from control theory. Lyapunov functions are dealt with in Chapter 0, various issues along them finite difference schemes are discussed in Chapter 10. Chapter 11 introduces viability tubes describing viability constraints changing with time. The final Chapters 12-14 deal with extensions to functional differential inclusions, partial differential inclusions and differential games.

The author takes great care in motivating and explaining the relevance of viability theory in general and of the contents of this book. In my opinion, mathematics will greatly benefit from more books written in this style.

Reviewer: F.Colonius (Augsburg)

### MSC:

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

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

93E03 | Stochastic systems in control theory (general) |