Lion and man game in the presence of a circular obstacle

In the lion and man game, a lion tries to capture a man who is as fast as the lion. We study a new version of this game which takes place in a Euclidean environment with a circular obstacle. We present a complete characterization of the game: for each player, we derive necessary and sufficient conditions for winning the game. Their (continuous time) strategies are constructed using techniques from differential games and arguments from geometry. Our main result is a decision algorithm which takes arbitrary initial positions as input, declares one of the players as the winner of the game and outputs a winning strategy for that player. We extend our approach to explicitly construct, in closed form, the decision boundary that partitions the arena into win and lose regions.