In this paper, we aim to recover the cost function that can explain given observations, for a class of constrained convex optimization problems. This problem in known as inverse optimal control and when studying it one should ask what criteria are minimized for given observations. The inverse optimal control was recently solved in an approximately optimal framework, meaning that the interest is in finding the proper criteria suitable to the system for which the decisions are approximately optimal. This method benefits of computational time efficiency and simplicity while solving the inverse optimal control problem. Applying a ’divide et impera’ principle the approximately optimal approach simplifies the initial optimization problem into easier ones. Being inspired by the approximately optimal approach, we focused on solving problems were systems and observation are both imperfect and uncertain. First, we apply this in order to verify its robustness while working with uncertain observations. Results show then that the method is sensitive to model uncertainties. Secondly, we propose a bounded-error approach to inverse optimal control ; where all uncertainty and disturbances acting on observation or modeling are assumed bounded but otherwise unknown. A set membership algorithm is then proposed that compute bounds on the set of criteria that make the uncertain observations optimal. We then show that the bounds computed for the cost function contains the actual solution.