Finite-horizon dynamic optimization of nonlinear systems in real time

A novel scheme for constructing and tracking the solution trajectories to regular, finite-horizon, deterministic optimal control problems with nonlinear dynamics is devised. The optimal control is obtained from the states and costates of Hamiltonian ODEs, integrated online. In the one-dimensional case the initial costate is found by successively solving two first-order, quasi-linear, partial differential equations, whose independent variables are the time-horizon duration T and the final penalty coefficient S. These PDEs should in general be integrated off-line, the solution rendering not only the missing initial condition sought in the particular (T,S)-situation, but additional information on the boundary values of the whole two-parameter family of control problems, which can be used for designing the definitive objective functional. Optimal trajectories for the model are then generated in real time and used as references to be followed by the physical system.  Numerical improvements are suggested for accurate integration of naturally unstable Hamiltonian dynamics, and strategies are proposed for tracking their results, in finite time or asymptotically, when perturbations in the state of the system appear. The whole procedure is tested in models arising in aero-navigation optimization.