Finite-Dimensional Characterization of Optimal Control Laws Over an Infinite Horizon for Nonlinear Systems

Infinite-horizon optimal control problems for nonlinear systems are considered. Due to the nonlinear and intrinsically
infinite-dimensional
nature of the task, solving such optimal control problems is challenging. In this article, an
exact finite-dimensional
characterization of the optimal solution over the entire horizon is proposed. This is obtained via the (static) minimization of a suitably defined function of (projected) trajectories of the underlying Hamiltonian dynamics on a hypersphere of fixed radius. The result is achieved in the spirit of the so-called
shooting methods
by introducing, via simultaneous forward/backward propagation, an
intermediate

shooting point
much closer to the origin, regardless of the actual initial state. A modified strategy allows one to determine an arbitrarily accurate approximate solution by means of standard gradient-descent algorithms over compact domains. Finally, to further increase the robustness of the control law, a receding-horizon architecture is envisioned by designing a sequence of shrinking hyperspheres. These aspects are illustrated by means of a benchmark numerical simulation.