E-raamat: Lyapunov Functions in Differential Games

Zhukovskiy (Russian Institute of Textile and Light Industry) examines the boundary of the theory of differential games, and develops a solution of dynamical games problems under uncertainty by means of the Bellman-Lyapunov function. The first of the two chapters describes the foundations of differential games under uncertainties and the notion of the vector guarantee. The second chapter proposes new guaranteeing solutions for differential linear quadratic games under uncertainty which are based on the equilibrium of objectives and counter-objectives as well as the active equilibrium. A detailed comparison is carried out with similar guaranteeing Nash equilibria. Annotation (c) Book News, Inc., Portland, OR (booknews.com)

A major step in differential games is determining an explicit form of the strategies of players who follow a certain optimality principle. To do this, the associated modification of Bellman dynamic programming problems has to be solved; for some differential games this could be Lyapunov functions whose "arsenal" has been supplied by stability theory. This approach, which combines dynamic programming and the Lyapunov function method, leads to coefficient criteria, or ratios of the game math model parameters with which optimal strategies of the players not only exist but their analytical form can be specified. In this book coefficient criteria are derived for numerous new and relevant problems in the theory of linear-quadratic multi-player differential games. Those criteria apply when the players formulate their strategies independently (non co-operative games) and use non-Nash equilibria or when the game model recognizes noise, perturbation and other uncertainties of which only their ranges are known (differential games under uncertainty). This text is useful for researchers, engineers and students of applied mathematics, control theory and the engineering sciences.