Lyapunov Functions in Differential Games
The book is devoted to the linear theory of differential games under uncertainty. It is supposed that nothing is known on their nature (i.e., the stochastic approach is ruled out). The presence of uncertainties moves the problem of the choice of strategy to the field of "multicriteria" dynamical problems. The results are mainly given for quadratic payoff functions, two players and non-cooperative games. The book is divided into two parts. The first one describes the foundations of differential games under uncertainties. The central role is played by the notion of a vector guarantee. Two approaches, based on the analogue of the vector saddle point and the vector maximin, are used for solving the multicriteria dynamical problems. The second part is devoted to the concept of equilibrium of objections and counter-objections, as well on the active equilibrium. The reader should have a basic knowledge of ordinary differential equations and optimization. For example, the dynamic programming approach (i.e., the Belman equation) and the method of Lyapunov functions are frequently used. Both parts end with results, comments on the history of ideas, and references (131 items). These are well oriented in works of the Russian and Ukrainian schools. Each part is also accompanied by exercises, with their solutions given at the end of the book.