A discrete-time Pontryagin maximum principle on matrix Lie groups

In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. The PMP provides first order necessary conditions for optimality; these necessary conditions typically yield two point boundary value problems, and these boundary value problems can then be solved to extract optimal control trajectories. Constrained optimal control problems for mechanical systems, in general, can only be solved numerically, and this motivates the need to derive discrete-time models that are accurate and preserve the non-flat manifold structures of the underlying continuous-time controlled systems. The PMPs for discrete-time systems evolving on Euclidean spaces are not readily applicable to discrete-time models evolving on non-flat manifolds. In this article we bridge this gap and establish a discrete-time PMP on matrix Lie groups. Our discrete-time models are derived via discrete mechanics, (a structure preserving discretization scheme) leading to the preservation of the underlying manifold under the dynamics, thereby resulting in greater numerical accuracy of our technique. This PMP caters to a class of constrained optimal control problems that includes point-wise state and control action constraints, and encompasses a large class of control problems that arise in various field of engineering and the applied sciences.