  In this paper, the solvability of the Inverse Optimal Control (IOC) problem
based on two existing minimum principal methods, is analysed. The aim of this
work is to answer the question regarding what kinds of trajectories, that is
depending on the initial conditions of the closed-loop system and system
dynamics, of the original optimal control problem, will result in the recovery
of the true weights of the reward function for both the soft and the
hard-constrained methods [1], [2]. Analytical conditions are provided which
allow to verify if a trajectory is sufficiently conditioned, that is, holds
sufficient information to recover the true weights of an optimal control
problem. It was found that the open-loop system of the original optimal problem
has a stronger influence on the solvability of the Inverse Optimal Control
problem for the hard-constrained method as compared to the soft-constrained
method. These analytical results were validated via simulation.
