Robust open-loop Nash equilibria in the noncooperative LQ game revisited

This paper reconsiders existence of worst‐case Nash equilibria in noncooperative multi‐player differential games, this, within an open‐loop information structure. We show that these equilibria can be obtained by determining the open‐loop Nash equilibria of an associated differential game with an additional initial state constraint. For the special case of linear‐quadratic differential games, we derive both necessary and sufficient conditions for solvability of the finite planning horizon problem. In particular, we demonstrate that, unlike in the standard linear‐quadratic differential game setting, uniqueness of equilibria may fail to hold. A both necessary and sufficient condition under which there is a unique equilibrium is provided. A sufficient existence condition for a unique equilibrium is derived in terms of a Riccati differential equation. Consequences for control policies are demonstrated in a simple debt stabilization game.