  In this note, we consider port-Hamiltonian structures in numerical optimal
control of ordinary differential equations. By introducing a novel class of
nonlinear monotone port-Hamiltonian (pH) systems, we show that the primal-dual
gradient method may be viewed as an infinite-dimensional nonlinear pH system.
The monotonicity and the particular block structure arising in the optimality
system is used to prove exponential stability of the dynamics towards its
equilibrium, which is a critical point of the first order optimality
conditions. Leveraging the port-based modeling, we propose an
optimization-based controller in a suboptimal receding horizon control fashion.
To this end, the primal-dual gradient based optimizer-dynamics is coupled to a
pH plant dynamics in a power-preserving manner. We show that the resulting
model is again monotone pH system and prove that the closed-loop exhibits local
exponential convergence towards the equilibrium.
