Iterative computation of noncooperative equilibria in nonzero-sum differential games with weakly coupled players

The authors study the Nash equilibria of a class of two-person nonlinear, deterministic differential games where the players are weakly coupled through the state equation and their objective functionals. The weak coupling is characterized in terms of a small perturbation parameter ε. With ε = 0, the problem decomposes into two dependent standard optimal control problems, while for ε ≠ 0, even though it is possible to derive the necessary and sufficient conditions to be satisfied by a Nash equilibrium solution, it is not always possible to construct such a solution. An iterative scheme is developed to obtain an approximate Nash solution when ε lies in a small interval around zero. Further, after requiring strong time consistency of the Nash equilibrium solution when at least one of the players uses dynamic information, the issue of existence and uniqueness of these solutions is studied for the cases when both players use the same information, either closed-loop or open-loop. It is also shown that even though the original problem is nonlinear, the higher (than zero'th) order terms in the Nash equilibria can be obtained as solutions to LQ (linear quadratic) optimal control problems, or static quadratic optimization problems.