Analytic solution for a class of linear quadratic open-loop Nash games

Abstract

A method for solving the asymmetric coupled Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic games is presented. The class of games studied here is one in which the state weighting matrices in player's cost functionals are proportional to each other. By writing in a special order the necessary conditions for open-loop Nash strategy, a matrix with specific properties is derived. These properties are then exploited to solve the two-point boundary-value problem. Some special cases are discussed and a simple example is given to illustrate the solution procedure.