Differential Games

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.