To play or not to play: a characterization of the marginal contribution of the opponent in a class of LQ differential games

A class of Linear Quadratic (LQ) finite-horizon differential games involving a Mayer-type cost on the state is studied with the aim of assessing the effect of the presence of an opponent. The contribution of the other player is quantitatively characterized by comparing the solutions of the underlying Riccati differential equations arising in the optimal control (in the absence of the opponent) and in the differential game. In the case of open-loop Nash equilibria, the contribution can be explicitly characterized, since closed-form solutions to the set of coupled asymmetric Riccati differential equations arising in the considered class of games can be computed. The construction of the underlying solutions hinges upon a time-varying change of coordinates, leading to constant equilibrium costate variables for each of the involved players. Moreover, in this setting, we provide further insights on the role of the costate variables similar to those arising in the context of optimal control problems. For feedback Nash equilibria a closed-form solution to the related coupled symmetric differential Riccati equations cannot be determined. Therefore an estimate of the solution is provided by relying on a functional approximation approach. It is shown that this is sufficient to characterize the contribution of the opponent also for feedback Nash Equilibria.