The sensitivity of terminal conditions of optimal control systems to parameter variations

The problem of sensitivity of specified terminal conditions to parameter variations is examined for three types of optimal systems: 1) open loop control 2) closed loop (feedback) control 3) closed loop control for trajectories neighboring an optimal open loop control trajectory. A method is proposed and illustrated for including the specification of sensitivity in open loop design. This is in contrast to analyzing sensitivity after the design has been completed. Open loop controls have some disadvantages from a sensitivity viewpoint. First of all, placing a constraint on the sensitivity will generally result in degradation of the performance. Furthermore, it may not even be possible to constrain the sensitivity to certain parameters. Closed loop controls would be expected to have superior sensitivity properties. Indeed it is proved that for a class of linear feedback systems, there is a finite range of parameter variations which have no effect on the terminal conditions. Furthermore, any bounded changes in some parameters have no effect on the terminal conditions. The results for the linear feedback laws are also applicable to neighboring optimal control systems at least in the case of small parameter variations.