Given x0, the control paths (i = 1, 2) obtained from (SM1) constitute an open-loop numerical solution of the couple’s effort problem. The scheme (SM2) allows us to compare the stabilizing solution with the unperturbed solution.
The latest 1D sort of brand new couple’s energy situation is noticed from inside the , whom proved the existence of a separate provider to your “lovebirds problem”, which is, offered a primary impact x(0) = x
One or two designs of one’s couple’s work problem are thought next. First, the newest 1D model of the problem is analysed. The newest opinions means brings here rewarding subservient suggestions with the (open-loop) control-theoretic procedures working in . After that, the research of your own dyadic (2D) types of the issue is addressed.
An equivalent factor thinking are used in both numerical knowledge. They are found in Table step 1. The fresh new electric and you may disutility functions utilized for the research, namely (14) are exactly the same as the those people considered when you look at the . This method is good to give their open-circle data of your condition. New energy and you will disutility features more than match the design requirement required in the last point. New mathematical performance showed inside section is actually sturdy relating to various needs of the design inputs.
Furthermore, for any initial feeling x0, the corresponding optimal trajectory (c ? (t), x ? (t)) converges towards the unique equilibrium of the following dynamical system, which is obtained from Pontryagin’s maximum principle, (15) The equilibrium is a saddle point, so the optimal trajectory lies on the stable manifold of the system (see or Theorem 1 in ).