  We establish a variety of results extending the well-known Pontryagin maximum
principle of optimal control to discrete-time optimal control problems posed on
smooth manifolds. These results are organized around a new theorem on critical
and approximate critical points for discrete-time geometric control systems. We
show that this theorem can be used to derive Lie group variational integrators
in Hamiltonian form; to establish a maximum principle for control problems in
the absence of state constraints; and to provide sufficient conditions for
exact penalization techniques in the presence of state or mixed constraints.
Exact penalization techniques are used to study sensitivity of the optimal
value function to constraint perturbations and to prove necessary optimality
conditions, including in the form of a maximum principle, for discrete-time
geometric control problems with state or mixed constraints.
