Differential Games

Abstract

We consider two players, Xavier and Yvette, and a differential game whose dynamics are described by

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaafa % qabeGabaaabaGaamyyaiaacMcadaGabaqaauaabeqaceaaaeaacaWG % PbGaaiykaiaayIW7ceWG4bGbauaacaGGOaGaamiDaiaacMcacqGH9a % qpcaWGMbGaaiikaiaadIhacaGGOaGaamiDaiaacMcacaGGSaGaamyE % aiaacIcacaWG0bGaaiykaiaacYcacaWG1bGaaiikaiaadshacaGGPa % GaaiykaaqaaiaadMgacaWGPbGaaiykaiaayIW7caWG1bGaaiikaiaa % dshacaGGPaGaeyicI4SaamyvaiaacIcacaWG4bGaaiikaiaadshaca % GGPaGaaiilaiaadMhacaGGOaGaamiDaiaacMcacaGGPaaaaaGaay5E % aaaabaGaamOyaiaacMcadaGabaqaauaabeqaceaaaeaacaWGPbGaai % ykaiaayIW7ceWG5bGbauaacaGGOaGaamiDaiaacMcacqGH9aqpcaWG % NbGaaiikaiaadIhacaGGOaGaamiDaiaacMcacaGGSaGaamyEaiaacI % cacaWG0bGaaiykaiaacYcacaWG2bGaaiikaiaadshacaGGPaGaaiyk % aaqaaiaadMgacaWGPbGaaiykaiaayIW7caWG2bGaaiikaiaadshaca % GGPaGaeyicI4SaamOvaiaacIcacaWG4bGaaiikaiaadshacaGGPaGa % aiilaiaadMhacaGGOaGaamiDaiaacMcacaGGPaaaaaGaay5Eaaaaaa % Gaay5Eaaaaaa!8BA5!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$\left\{ {\begin{array}{*{20}{c}} {a)\left\{ {\begin{array}{*{20}{c}} {i){\kern 1pt} x'(t) = f(x(t),y(t),u(t))} \\ {ii){\kern 1pt} u(t) \in U(x(t),y(t))} \end{array}} \right.} \\ {b)\left\{ {\begin{array}{*{20}{c}} {i){\kern 1pt} y'(t) = g(x(t),y(t),v(t))} \\ {ii){\kern 1pt} v(t) \in V(x(t),y(t))} \end{array}} \right.} \end{array}} \right.$$

where u, v, the controls, are regarded as strategies used by the players to govern the evolution of the states x, y of the game.