On the open-loop Nash equilibrium in LQ-games

In this paper we consider open-loop Nash equilibria of the linear-quadratic (LQ) differential game. We present both necessary and sufficient conditions for existence of a unique solution for the finite-planning horizon case, and show that there exist situations where the set of associated Riccati differential equations has no solution, whereas the problem does have an equilibrium. The pursued analyses allows a simple study of convergence of the equilibrium strategy if the planning horizon expands. Conditions are given under which this strategy converges. A detailed study of the infinite planning horizon case is given. In particular we show that, in general, this problem has no unique equilibrium. Furthermore, we show that the limit of the above mentioned converged strategy may not be an equilibrium for the infinite planning horizon problem. Particular attention is paid to computational aspects and the scalar case. (C) 1998 Elsevier Science B.V. All rights reserved.