On a Probabilistic Approach for Inverse Data-Driven Optimal Control

We consider the problem of estimating the possibly non-convex cost of an agent by observing its interactions with a nonlinear, non-stationary and stochastic environment. For this inverse problem, we give a result that allows to estimate the cost by solving a convex optimization problem. To obtain this result we also tackle a forward problem. This leads to the formulation of a finite-horizon optimal control problem for which we show convexity and find the optimal solution. Our approach leverages certain probabilistic descriptions that can be obtained both from data and/or from first-principles. The effectiveness of our results, which are turned in an algorithm, is illustrated via simulations on the problem of estimating the cost of an agent that is stabilizing the unstable equilibrium of a pendulum.