A differential dynamic programming (DDP)-based framework for inverse reinforcement learning (IRL) is introduced to recover the parameters in the cost function, system dynamics, and constraints from demonstrations. Different from existing work, where DDP was usually used for the inner forward problem, our proposed framework uses it to efficiently compute the gradient required in the outer inverse problem with equality and inequality constraints. The equivalence between the proposed and existing methods based on Pontryagin’s Maximum Principle (PMP) is established. More importantly, using this DDPbased IRL with an open-loop loss function, a closed-loop IRL framework is presented. In this framework, a loss function is proposed to capture the closed-loop nature of demonstrations. It is shown to be better than the commonly used open-loop loss function. We show that the closed-loop IRL framework reduces to a constrained inverse optimal control problem under certain assumptions. Under these assumptions and a rank condition, it is proven that the learning parameters can be recovered from the demonstration data. The proposed framework is extensively evaluated through four numerical robot examples and one realworld quadrotor system. The experiments validate the theoretical results and illustrate the practical relevance of the approach.