  The Inverse Optimal Control (IOC) problem is a structured system
identification problem that aims to identify the underlying objective function
based on observed optimal trajectories. This provides a data-driven way to
model experts' behavior. In this paper, we consider the case of discrete-time
finite-horizon linear-quadratic problems where: the quadratic cost term in the
objective is not necessarily positive semi-definite; the planning horizon is a
random variable; we have both process noise and observation noise; the dynamics
can have a drift term; and where we can have a linear cost term in the
objective. In this setting, we first formulate the necessary and sufficient
conditions for when the forward optimal control problem is solvable. Next, we
show that the corresponding IOC problem is identifiable. Using the conditions
for existence of an optimum of the forward problem, we then formulate an
estimator for the parameters in the objective function of the forward problem
as the globally optimal solution to a convex optimization problem, and prove
that the estimator is statistical consistent. Finally, the performance of the
algorithm is demonstrated on two numerical examples.
