Chapter 2 A DIRECT METHOD FOR OPEN-LOOP DYNAMIC GAMES FOR AFFINE CONTROL SYSTEMS

mann’s direct method, first presented for problems in the calculus of variations in Leitmann (1967), for open-loop dynamic games in Dock-ner and Leitmann (2001) were given. In these papers each player has its own state which it controls with its own control inputs. That is, there is a state equation for each player. However, many applications involve the players competing for a single resource (e.g., two countries competing for a single species of fish). In this note we investigate the utility of the direct method for a class of games whose dynamics are described by a single equation for which the state dynamics are affine in the players strategies. An illustrative example is also presented 1. The direct method In Carlson and Leitmann (2004) a direct method for finding open-loop Nash equilibria for a class of differential N-player games is presented. A particular case included in this study concerns the situation in which the j-th player’s dynamics at any time t ∈ [t0, tf] is a vector-valued functio