A common assumption in physiology about human motion is that the realized movements are done in an optimal way. The problem of recovering of the optimality principle leads to the inverse optimal control problem. Formally, in the inverse optimal control problem we should find a cost function such that under the known dynamical constraint the observed trajectories are minimizing for such cost. In this paper we analyze the inverse problem in the case of finite horizon linear-quadratic problem. In particular, we treat the injectivity question, i.e. whether the cost corresponding to the given data is unique, and we propose a cost reconstruction algorithm. In our approach we define the canonical class on which the inverse problem is either unique or admit a special structure, which can be used in cost reconstruction.