This paper is devoted to the general problem of reconstructing the cost from the observation of trajectories, in a problem of optimal control. It is motivated by the following applied problem, concerning HALE drones: one would like them to decide by themselves for their trajectories, and to behave at least as a good human pilot. This applied question is very similar to the problem of determining what is minimized in human locomotion. These starting points are the reasons for the particular classes...

The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form: ${\displaystyle \underset{{\displaystyle (u,v)\in {𝒰}_{ad}}}{inf}{\int }_0^{1}f\left(t,u\left({\theta }_v\left(t\right)\right),u^{\text{'}}\left(t\right),v\left(t\right)\right)\mathrm{d}t}$, (1) where ${𝒰}_{ad}$ is a set of admissible controls and ${\theta }_v$ is the solution of the following equation: $\{\frac{\mathrm{d}\theta \left(t\right)}{\mathrm{d}t}=g(t,\theta \left(t\right),v\left(t\right)),t\in [0,1]$ ; ${\displaystyle \theta \left(0\right)={\theta }_0,\theta \left(t\right)\in [0,1]\forall t}$. (2). The results are nonlocal and new.