  Discrete control systems, as considered here, refer to the control theory of
discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are
based on a discrete variational principle, and are part of the broader field of
geometric integration. Geometric integrators are numerical integration methods
that preserve geometric properties of continuous systems, such as conservation
of the symplectic form, momentum, and energy. They also guarantee that the
discrete flow remains on the manifold on which the continuous system evolves,
an important property in the case of rigid-body dynamics.
  In nonlinear control, one typically relies on differential geometric and
dynamical systems techniques to prove properties such as stability,
controllability, and optimality. More generally, the geometric structure of
such systems plays a critical role in the nonlinear analysis of the
corresponding control problems. Despite the critical role of geometry and
mechanics in the analysis of nonlinear control systems, nonlinear control
algorithms have typically been implemented using numerical schemes that ignore
the underlying geometry.
  The field of discrete control system aims to address this deficiency by
restricting the approximation to choice of a discrete-time model, and
developing an associated control theory that does not introduce any additional
approximation. In particular, this involves the construction of a control
theory for discrete-time models based on geometric integrators that yields
numerical implementations of nonlinear and geometric control algorithms that
preserve the crucial underlying geometric structure.
