Inverse optimal control problem emerges in different practical applications, where the goal is to design a cost function in order to approximate given optimal strategies of an expert. Typical application is in robotics for generation of human motions. In this paper we analyze a general class of non autonomous inverse linear quadratic problems. This class of problems is of particular interest because it arises as a linearization of a nonlinear problem around an optimal trajectory. The addressed questions are the injectivity of the inverse problem and the reconstruction. We show that the nonlinear problem admits the same characterization of the injectivity as the autonomous one. In the autonomous case we show moreover that the injectivity property is generic in the considered class. We also provide a numerical test of the reconstruction algorithm in the autonomous setting.