The open-loop Nash equilibrium in LQ-games revisited

In this paper we reconsider the conditions under which the finite-planning-horizon linear- quadratic differential game has an open-loop Nash equilibrium solution. Both necessary and sufficient conditions are presented for the existence of a unique solution in terms of an invertibility condition on a matrix. Moreover, we show that the often encountered solvability conditions stated in terms of Riccati equations are in general not correct. In an example we show that existence of a solution of the associated Riccati-type differential equations may fail to exist whereas an open-loop Nash equilibrium solution exists. The scalar case is studied in more detail, and we show that solvability of the associated Riccati equations is in that case both necessary and sufficient. Furthermore we consider convergence properties of the open-loop Nash equilibrium solution if the planning horizon is extended to infinity. To study this aspect we consider the existence of real solutions of the associated algebraic Riccati equation in detail and show how all solutions can be easily calculated from the eigenstructure of a matrix.