  We present the concept of a Generalized Feedback Nash Equilibrium (GFNE) in
dynamic games, extending the Feedback Nash Equilibrium concept to games in
which players are subject to state and input constraints. We formalize
necessary and sufficient conditions for (local) GFNE solutions at the
trajectory level, which enable the development of efficient numerical methods
for their computation. Specifically, we propose a Newton-style method for
finding game trajectories which satisfy the necessary conditions, which can
then be checked against the sufficiency conditions. We show that the evaluation
of the necessary conditions in general requires computing a series of nested,
implicitly-defined derivatives, which quickly becomes intractable. To this end,
we introduce an approximation to the necessary conditions which is amenable to
efficient evaluation, and in turn, computation of solutions. We term the
solutions to the approximate necessary conditions Generalized Feedback Quasi
Nash Equilibria (GFQNE), and we introduce numerical methods for their
computation. In particular, we develop a Sequential Linear-Quadratic Game
approach, in which a locally approximate LQ game is solved at each iteration.
The development of this method relies on the ability to compute a GFNE to
inequality- and equality-constrained LQ games, and therefore specific methods
for the solution of these special cases are developed in detail. We demonstrate
the effectiveness of the proposed solution approach on a dynamic game arising
in an autonomous driving application.
