1. Game-Theoretical Control.- 1.1 Minimax Control Problems.- 1.2 Equations of Motion.- 1.3 Positional Strategies.- 1.4 Quality Criterion.- 1.5 Differential Games.- 2. Differential Game of Approach-Evasion.- 2.1 Game-Theoretical Problems of Approach and Evasion.- 2.2 Stable Bridges.- 2.3 Local Estimates.- 2.4 Extremal Strategies.- 2.5 Construction of Stable Bridges.- 3. Existence of the Value for a Positional Differential Game.- 3.1 A Differential Game with a Fixed Termination Time.- 3.2 Differential Games of Pursuit-Evasion.- 3.3 Differential Games with a Bounded Duration.- 3.4 Value Function.- 3.5 Examples.- 4. Dynamic Programming.- 4.1 The Main Equation for a Smooth Value Function.- 4.2 Properties of Value Functions.- 4.3 Directional Differentiability.- 4.4 Generalization of the Main Equation.- 4.5 Examples.- 5. Extremal Aiming.- 5.1 Programmed Maximin and Programmed Absorption.- 5.2 Extremal Aiming in Linear Controlled Systems.- 5.3 Extremal Aiming in Problems of Pursuit and Evasion.- 5.4 Regularity Conditions for a Programmed Maximin.- 5.5 Differentiability of the Value Function and Regularity Condition for the Programmed Maximin.- 6. Extremal Aiming for Nonlinear Differential Games.- 6.1 Generalized Programmed Controls.- 6.2 Programmed Maximin.- 6.3 The Minimum Principle and Maximin Rule.- 6.4 Directional Derivatives of Functions of a Programmed Maximin.- 6.5 Stability Conditions for the Programmed Maximin.- 7. Prior Stable Sets.- 7.1 Prior Stable Path.- 7.2 Stable Integral Manifolds.- 7.3 Prior Stable Sets for a Linear System.- 7.4 Comparison of Controlled Systems.- 8. Qualitative Problems in the Theory of Differential Games.- 8.1 The Problem of Stability of Solutions.- 8.2 Guidance Control.- 8.3 Stabilization of Solutions of Game-Theoretical Control Problems.- 8.4 Evasion on an Infinite Time Interval.- 8.5 Universal Optimal Strategies.- 9. Mixed Strategies in Differential Games.- 9.1 Deterministic and Stochastic Controls.- 9.2 Mixed Strategies.- 9.3 Dynamic Programming, Extremal Aiming, and A Priori Stable Sets.- 9.4 Approximation of Mixed Strategies.- 9.5 Stochastic Control with Guidance.- 10. Lower and Upper Differential Games.- 10.1 Counterstrategies.- 10.2 Alternatives.- 10.3 Efficient Solutions of Lower and Upper Differential Games.- 10.4 Control Procedures with Guidance.- 10.5 Unification of Differential Games.- 11. Differential-Functional Games.- 1.1 Strategies with Complete Memory.- 1.2 Stable Functionals.- 1.3 The Saddle Point.- 1.4 Properties of Value Functionals.- A.l. Semi-Continuous Functions.- A.2. Convex Functions.- A.3. Multivalued Mappings.- A.4. Differential Inclusions.
This book is devoted to an investigation of control problems which can be described by ordinary differential equations and be expressed in terms of game theoretical notions. In these terms, a strategy is a control based on the feedback principle which will assure a definite equality for the controlled process which is subject to uncertain factors such as a move or a controlling action of the opponent. Game Theoretical Control Problems contains definitions and formalizations of differential games, existence for equilibrium and extensive discussions of optimal strategies. Formal definitions and statements are accompanied by suitable motivations and discussions of computational algorithms. The book is addessed to mathematicians, engineers, economists and other users of control theoretical and game theoretical notions.
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