  Designing controllers to generate various trajectories has been studied for
years, while recently, recovering an optimal controller from trajectories
receives increasing attention. In this paper, we reveal that the inherent
linear quadratic regulator (LQR) problem of a moving agent can be reconstructed
based on its trajectory observations only, which enables one to learn the
optimal control law of the agent autonomously. Specifically, the reconstruction
of the optimization problem requires estimation of three unknown parameters
including the target state, weighting matrices in the objective function and
the control horizon. Our algorithm considers two types of objective function
settings and identifies the weighting matrices with proposed novel inverse
optimal control methods, providing the well-posedness and identifiability
proof. We obtain the optimal estimate of the control horizon using binary
search and finally reconstruct the LQR problem with above estimates. The
strength of learning control law with optimization problem recovery lies in
less computation consumption and strong generalization ability. We apply our
algorithm to the future control input prediction and the discrepancy loss is
further derived. Numerical simulations and hardware experiments on a
self-designed robot platform illustrate the effectiveness of our work.
