##
**Game-theoretical control problems.
Transl. from the Russian by Samuel Kotz.**
*(English)*
Zbl 0649.90101

Springer Series in Soviet Mathematics. New York etc.: Springer-Verlag. XI, 517 p.; DM 248.00 (1988).

This is an English version of the authors’ monograph “Positional differential games” (1974; Zbl 0298.90067). The Russian variant was based on results obtained by the authors and their collaborators in the early 1970s. For the English translation the monograph was substantially revised in order to reflect results of intensive investigations up to today.

The book deals with conflict-controlled processes described by ordinary differential equations. It is necessary to find a feedback control action, i.e. positional strategy, which ensures a definite quality for the controlled process under the presence of uncertain factors. Uncertain factors may be considered as unknown disturbances or control actions of the opposite side. Methods for solving such problems are presented.

The authors’ conception of a strictly formalized positional differential game turned out to be quite viable and continues to serve as a sound basis for new developments. In the reviewed book this conception is based on the notion of a guaranteed result. By solving minimax and maximin problems the players estimate the quality of their strategies by a guaranteed result. The notions of the value of the game and of a saddle point are introduced next. These are basic notions which combine the two problems with guaranteed result into a differential game.

An investigation of differential games with various types of quality criteria may be reduced to the corresponding game of approach-evasion. For investigating approach-evasion games notions of stable bridges and strategies extremal to these bridges are derived and a theorem on an alternative is proved. The proof is new, more compact and more complete. The theorem on the alternative shows that the approach-evasion game possesses an equilibrium situation and, therefore, different types of differential games have a value for any choice of the initial position. Recursive and iterative constructions for determining the value function are described.

A number of new sections devoted to properties of the value function are presented. It is shown that the value function is continuous or satisfies the Lipschitz condition. A generalization of the method of dynamic programming is obtained in a form of a group of necessary as well as sufficient conditions to be satisfied by the directional derivatives of the value function. At the points of differentiability of the value function these conditions become the well-known first order partial differential equation of the Hamilton-Jacobi type. If the value is piecewise smooth then these conditions are calculated easily and may be used for verifying a hypothesis about the value.

The main attention is paid to tracing of relatively simple computational algorithms which are based on auxiliary programme constructions. So- called regularity conditions are derived under which the value of the game coincides with the programmed maximin. The new result is that the differentiable value of the game coincides with the programmed maximin. In regular cases determination of the value and optimal strategies is reduced to problems of mathematical programming. Such an approach is well-known as the method of extremal aiming. As in the Russian edition a substantiation and investigation of the method of extremal aiming is given for linear and nonlinear differential games. We single out here a new interesting result: all games which have identical Hamiltonian are equivalent and have the same value. Consequently, the initial system may be exchanged by a certain simplified auxiliary system.

All notions and constructions are transferred to differential-functional games when the dynamics depends on the motion realized at the current time. In particular, games whose dynamics depend on various types of lags belongs to this class. For certain types of such games there is no equilibrium situation in positional strategies and under suitable conditions an equilibrium situation may be obtained by broadening the information available to the players, i.e. by assuming that the players construct their controls knowing the motion realized at the current time. For differential-functional games the stable bridges are defined in the space of continuous functions and solutions of games are given with the aid of strategies extremal to stable bridges. It is proved that extremal strategies form a saddle point and certain properties of value functionals are investigated. This is a new material also included in the English edition.

Plenty of illustrating examples are given and, moreover, some of them are of independent interest. Thus, the book summarizes and unifies the important contributions to the theory of positional differential games made by the authors and their collaborators up to today. The proofs are substantially revised and improved. Because of methodical improvements the material is presented more compact and in a more understandable form.

The monograph presents general and fundamental results in the modern theory of differential games which are a useful foundation for various applications in practice. Undoubtedly, it will win recognition by specialists and get an important place in the scientific literature.

The book deals with conflict-controlled processes described by ordinary differential equations. It is necessary to find a feedback control action, i.e. positional strategy, which ensures a definite quality for the controlled process under the presence of uncertain factors. Uncertain factors may be considered as unknown disturbances or control actions of the opposite side. Methods for solving such problems are presented.

The authors’ conception of a strictly formalized positional differential game turned out to be quite viable and continues to serve as a sound basis for new developments. In the reviewed book this conception is based on the notion of a guaranteed result. By solving minimax and maximin problems the players estimate the quality of their strategies by a guaranteed result. The notions of the value of the game and of a saddle point are introduced next. These are basic notions which combine the two problems with guaranteed result into a differential game.

An investigation of differential games with various types of quality criteria may be reduced to the corresponding game of approach-evasion. For investigating approach-evasion games notions of stable bridges and strategies extremal to these bridges are derived and a theorem on an alternative is proved. The proof is new, more compact and more complete. The theorem on the alternative shows that the approach-evasion game possesses an equilibrium situation and, therefore, different types of differential games have a value for any choice of the initial position. Recursive and iterative constructions for determining the value function are described.

A number of new sections devoted to properties of the value function are presented. It is shown that the value function is continuous or satisfies the Lipschitz condition. A generalization of the method of dynamic programming is obtained in a form of a group of necessary as well as sufficient conditions to be satisfied by the directional derivatives of the value function. At the points of differentiability of the value function these conditions become the well-known first order partial differential equation of the Hamilton-Jacobi type. If the value is piecewise smooth then these conditions are calculated easily and may be used for verifying a hypothesis about the value.

The main attention is paid to tracing of relatively simple computational algorithms which are based on auxiliary programme constructions. So- called regularity conditions are derived under which the value of the game coincides with the programmed maximin. The new result is that the differentiable value of the game coincides with the programmed maximin. In regular cases determination of the value and optimal strategies is reduced to problems of mathematical programming. Such an approach is well-known as the method of extremal aiming. As in the Russian edition a substantiation and investigation of the method of extremal aiming is given for linear and nonlinear differential games. We single out here a new interesting result: all games which have identical Hamiltonian are equivalent and have the same value. Consequently, the initial system may be exchanged by a certain simplified auxiliary system.

All notions and constructions are transferred to differential-functional games when the dynamics depends on the motion realized at the current time. In particular, games whose dynamics depend on various types of lags belongs to this class. For certain types of such games there is no equilibrium situation in positional strategies and under suitable conditions an equilibrium situation may be obtained by broadening the information available to the players, i.e. by assuming that the players construct their controls knowing the motion realized at the current time. For differential-functional games the stable bridges are defined in the space of continuous functions and solutions of games are given with the aid of strategies extremal to stable bridges. It is proved that extremal strategies form a saddle point and certain properties of value functionals are investigated. This is a new material also included in the English edition.

Plenty of illustrating examples are given and, moreover, some of them are of independent interest. Thus, the book summarizes and unifies the important contributions to the theory of positional differential games made by the authors and their collaborators up to today. The proofs are substantially revised and improved. Because of methodical improvements the material is presented more compact and in a more understandable form.

The monograph presents general and fundamental results in the modern theory of differential games which are a useful foundation for various applications in practice. Undoubtedly, it will win recognition by specialists and get an important place in the scientific literature.

Reviewer: E.Al’brekht

### MSC:

91A23 | Differential games (aspects of game theory) |

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |

49J35 | Existence of solutions for minimax problems |

91A99 | Game theory |

91A24 | Positional games (pursuit and evasion, etc.) |

90C39 | Dynamic programming |

49L20 | Dynamic programming in optimal control and differential games |