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.
Table of Contents
The Simplist Concepts and Examples. Some Concepts in the Theory of Differential Games Under Uncertainty. Game Problems in Mechanical and Economical Systems. Vector-Valued Guarantees. Vector-Valued Guarantees Can Exist or Not. Converse Problem. Equilibrium of Nash Under Uncertainty. Equilibrium of Threats and Counterthreats Under Uncertainty. Singularities of the Nash Equilibrium. Formalization and Properties Unimprovable Equilibriums. Comparison with Nash Equilibrium. Formalization of Unimprovable Equilibriums in Differential Game. Auxiliary Propositions. Sufficient Conditions for the Saddle-Point Analogy. Unimprovable Guaranteeing Equilibriums (Vector-Valued Max/min Analogy). Active Equilibrium Under Uncertainty. Berge Equilibrium Under Uncertainty. Formalization of the Solutions. Games with Separable Payoff Function. Strictly Convex Games Under Uncertainty. Properties of Berge Equilibrium. Linear-Quadratic Differential Game of Three Persons Under Uncertainty. Appendix 1: From the Theory of Differential Equations. Appendix 2: From the Theory of Quadratic Forms. Appendix 3: From the Theory of Mathematical Programming. Appendix 4: Auxiliary Propositions.