Differential Games: Theory and Methods for Solving Game Problems with Singular Surfaces

ISBN-13: 9781447120674 / Angielski / Miękka / 2011 / 242 str.

Joseph Lewin

Opis

Differential games theory is the most appropriate discipline for the modelling and analysis of real life conflict problems.
The theory of differential games is here treated with an emphasis on the construction of solutions to actual problems with singular surfaces. The reader is provided with the knowledge necessary to put the theory of differential games into practice.

Spis treści

1 A Preview Example.- 1.1 Introduction.- 1.2 A Simple Differential Game.- 1.3 Preliminary Analysis.- 1.4 A Heuristic Solution.- 1.5 Problems.- 2 The Vocabulary For Differential Games.- 2.1 Introduction.- 2.2 The State Vector and the Game-Set.- 2.3 The Equations of Motion.- 2.4 Termigation of a Differential Game.- 2.5 Plays.- 2.6 Outcomes.- 2.7 Strategies.- 2.7.1 Decisions and Information.- 2.7.2 Realizations of Strategies.- 2.7.3 Strategies that Guarantee Nontermination.- 2.7.4 Strategies that Guarantee Termination.- 2.7.5 Admissibility of Strategies.- 2.8 Problems.- 3 The Solution Concept.- 3.1 Introduction.- 3.2 The Solution Quintet.- 3.3 The Extended Solution Concept.- 3.4 Problems.- 4 Semipermeability of Surfaces.- 4.1 Introduction.- 4.2 Smooth Semipermeable Surfaces.- 4.3 Semipermeability of Composite Surfaces.- 4.3.1 Leaking Corners.- 4.3.2 A Modified Definition of Semipermeability.- 4.4 Problems.- 5 Necessary Conditions.- 5.1 Introduction.- 5.2 Properties of the Target Set.- 5.2.1 Partitioning of the Target Set.- 5.2.2 The Relation Between J(x) and G(x).- 5.3 Semipermeability of the Boundary of the Escape Set F.- 5.4 Properties of Optimal Trajectories.- 5.4.1 Principle of Optimality (weak).- 5.4.2 Continuity Of The Value Function.- 5.4.3 j(x) Along Optimal Trajectories.- 5.4.4 The Hamiltonian on Optimal Trajectories.- 5.5 The Isaacs Equations.- 5.5.1 Semi-Local Deviations From Optimality.- 5.5.2 The Isaacs Main Equations.- 5.5.3 The hodograph representations of ME1.- 5.5.4 The Viscosity Form of Isaacs Equations.- 5.6 The Adjoint Equations.- 5.6.1 The Retro Time Form of the Adjoint Equations.- 5.7 Problems.- 6 Sufficient Conditions.- 6.1 Introduction.- 6.2 The Sufficiency Theorem.- 6.3 Validity of Partial Solutions.- 6.4 Estimatioms of the Value Function.- 6.5 Problems.- 7 Regular Construction.- 7.1 Introduction.- 7.2 The Regular Procedure.- 7.2.1 Partitioning the Target-Set.- 7.2.2 Candidate Optimal Control Laws.- 7.2.3 Retro-Integration of the Adjoint Equations.- 7.2.4 Properties of the Manifolds of Candidate Optimal Trajectories.- 7.3 Examples.- 7.4 Linear Quadratic Games.- 7.4.1 Introduction.- 7.4.2 LQG with Fixed Duration and Unbounded Controls.- 7.4.3 Infinite Horizon Linear Quadratic Games.- 7.4.4 LQG and Controller Design.- 7.5 Problems.- 8 Construction of SPS.- 8.1 Introduction.- 8.2 Construction of Semipermeable Surfaces.- 8.2.1 The Regular Construction.- 8.2.2 Semipermeability of the Constructed Manifold.- 8.3 Examples.- 8.4 Problems.- 9 A Topography of the Value Map.- 9.1 Introduction.- 9.2 Barriers and Safe Contact.- 9.2.1 Barriers.- 9.2.2 State Costraints.- 9.2.3 Safe Contact.- 9.2.4 The Tributaries.- 9.3 Switch Surfaces.- 9.4 Dispersal Surfaces.- 9.5 Universal and Focal Surfaces.- 9.5.1 General characterization.- 9.5.2 Universal Surfaces.- 9.5.3 Focal Surfaces.- 9.6 Corner Surfaces.- 9.6.1 General characterization.- 9.6.2 Equivocal Surfaces.- 9.6.3 Switch Envelopes.- 10 Necessary Conditions (Singular).- 10.1 Introduction.- 10.2 The Projection Lemma.- 10.3 Open Barriers.- 10.4 Isaacs Equations for Singular Arcs.- 10.4.1 The Hamiltonian on Singular Surfaces.- 10.4.2 Isaacs Theorems for Singular Arcs.- 10.4.3 Hamiltonians on Seams.- 10.5 Junctions to Singular Arcs.- 10.5.1 Controls Along Singular Arcs.- 10.5.2 The Junction Conditions.- 10.6 Adjoint Equations for Singular Arcs.- 10.7 Properties of Regular Switch Surfaces.- 10.8 The Chatter Equivalent of Singular Arcs.- 10.8.1 Introduction.- 10.8.2 Singular Arcs with Tributaries Joining Transversely.- 10.8.3 Singular Arcs with Tributaries Joining Tangentially.- 10.9 Sufficient conditions.- 10.l0 Problems.- 11 Dispersal Surfaces.- 11.1 introduction.- 11.2 Region of Multiple Choices.- 11.3 Characterization of Dispersal Surfaces.- 11.4 Examples.- 11.5 Problems.- 12 Singular Arcs of Safe Contact.- 12.1 Introduction.- 12.2 Characterization of Safe Contact.- 12.3 Construction of Safe Contact Arcs.- 12.3.1 Introduction.- 12.3.2 Safe Contact with Tangential Junctions.- 12.3.3 Safe Contact with Transversal Junctions.- 12.4 Examples.- 12.5 Problems.- 13 Universal and Focal Surfaces.- 13.1 Introduction.- 13.2 Characterization of Universal Surfaces.- 13.3 Examples.- 13.3.1 The Chatter Equivalent.- 13.4 Characterization of Focal Surfaces.- 13.5 Construction of Focal Surfaces.- 13.6 An Example of a Focal Surface.- 13.6.1 The Chatter Equivalent.- 13.7 Problems.- 14 Corner Surfaces.- 14.1 Introduction.- 14.2 Characterization of Corner Surfaces.- 14.3 The Switch Envelope.- 14.4 Chatter Equivalent of SE.- 14.5 The Equivocal Surface.- 14.6 Chatter Equivalent of ES.- 14.7 Problems.- 15 The Envelope Barrier.- 15.1 Introduction.- 15.2 The Envelope Barrier.- 15.2.1 Dominated Surfaces.- 15.2.2 Characterization of Envelope Barriers.- 15.3 Examples.

O autorze

Differential Games is particularly intended for those who want to apply the theory of differential games to real applications. It therefore includes numerous exercises and worked examples to help the reader gain practice and experience. It can be used for a graduate course of self study.