Differential Hybrid Games (CMU-CS-14-102)

This paper introduces differential hybrid games, which combine differential games with hybrid games. In both kinds of games, two players interact with continuous dynamics. The difference is that hybrid games also provide all the features of hybrid systems and discrete games, but only deterministic differential equations. Differential games, instead, provide differential equations with input by both players, but not the luxury of hybrid games, such as mode switches and discrete or alternating interaction. This paper augments differential game logic with modalities for the combined dynamics of differential hybrid games. It shows how hybrid games subsume differential games and introduces differential game invariants and differential game variants for proving properties of differential games inductively. This material is based upon work supported by the National Science Foundation under NSF CAREER Award CNS-1054246. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of any sponsoring institution or government.

[1]  Rufus Isaacs,et al.  Differential Games , 1965 .

[2]  A. Friedman.  Differential games , 1971 .

[3]  R. Elliott,et al.  Cauchy problems for certain Isaacs-Bellman equations and games of survival , 1974 .

[4]  P. Souganidis,et al.  Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations. , 1983 .

[5]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[6]  L. Evans,et al.  Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls , 1984 .

[7]  P. Souganidis.  Approximation schemes for viscosity solutions of Hamilton-Jacobi equations , 1985 .

[8]  A. I. Subbotin,et al.  Game-Theoretical Control Problems , 1987 .

[9]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[10]  Leon A. Petrosjan,et al.  Differential Games of Pursuit , 1993 .

[11]  P. Cardaliaguet.  A differential game with two players and one target , 1994 .

[12]  Anil Nerode,et al.  Viability in Hybrid Systems , 1995, Theor. Comput. Sci..

[13]  Anil Nerode,et al.  Hybrid System Games: Extraction of Control Automata with Small Topologies , 1996, Hybrid Systems.

[14]  Thomas A. Henzinger,et al.  The theory of hybrid automata , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[15]  Duvsan Repovvs,et al.  Continuous Selections of Multivalued Mappings , 1998, 1401.2257.

[16]  Alain Rapaport,et al.  Characterization of Barriers of Differential Games , 1998 .

[17]  Thomas A. Henzinger,et al.  Rectangular Hybrid Games , 1999, CONCUR.

[18]  T. Başar,et al.  Stochastic and differential games : theory and numerical methods , 1999 .

[19]  J. Lygeros,et al.  A game theoretic approach to controller design for hybrid systems , 2000, Proceedings of the IEEE.

[20]  D. Marker.  Model theory : an introduction , 2002 .

[21]  Oana-Silvia Serea,et al.  Discontinuous differential games and control systems with supremum cost , 2002 .

[22]  G. Sacks.  A DECISION METHOD FOR ELEMENTARY ALGEBRA AND GEOMETRY , 2003 .

[23]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[24]  Patrick Saint-Pierre.  Viable Capture Basin for Studying differential and Hybrid Games: Application to Finance , 2004, IGTR.

[25]  Alexandre M. Bayen,et al.  A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games , 2005, IEEE Transactions on Automatic Control.

[26]  Mythily Ramaswamy,et al.  Zero-Sum Differential Games Involving Hybrid Controls , 2006 .

[27]  Alexandre M. Bayen,et al.  Viability-Based Computations of Solutions to the Hamilton-Jacobi-Bellman Equation , 2007, HSCC.

[28]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[29]  Yan Gao,et al.  On the Reachability Problem for Uncertain Hybrid Systems , 2007, IEEE Transactions on Automatic Control.

[30]  Marc Quincampoix,et al.  Differential Games Through Viability Theory: Old and Recent Results , 2007 .

[31]  André Platzer,et al.  Differential Dynamic Logic for Hybrid Systems , 2008, Journal of Automated Reasoning.

[32]  Otomar Hájek,et al.  Pursuit Games: An Introduction to the Theory and Applications of Differential Games of Pursuit and Evasion , 2008 .

[33]  Edmund M. Clarke,et al.  Computing differential invariants of hybrid systems as fixedpoints , 2008, Formal Methods Syst. Des..

[34]  Patricia Bouyer,et al.  O-Minimal Hybrid Reachability Games , 2009, Log. Methods Comput. Sci..

[35]  André Platzer,et al.  Differential-algebraic Dynamic Logic for Differential-algebraic Programs , 2010, J. Log. Comput..

[36]  R. Newcomb.  VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS , 2010 .

[37]  Alberto Bressan,et al.  Viscosity Solutions of Hamilton-Jacobi Equations and Optimal Control Problems , 2011 .

[38]  Lars Grüne,et al.  Differential Games and Zubov's Method , 2011, SIAM J. Control. Optim..

[39]  André Platzer,et al.  The Structure of Differential Invariants and Differential Cut Elimination , 2011, Log. Methods Comput. Sci..

[40]  Mahesh Viswanathan,et al.  Specifications for decidable hybrid games , 2011, Theor. Comput. Sci..

[41]  André Platzer,et al.  Logics of Dynamical Systems , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[42]  André Platzer,et al.  The Complete Proof Theory of Hybrid Systems , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[43]  G. Barles.  An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications , 2013 .

[44]  André Platzer,et al.  Differential Game Logic , 2014, ACM Trans. Comput. Log..