Data-Driven Optimal Control of Nonlinear Dynamics Under Safety Constraints

This letter considers the optimal control problem of nonlinear systems under safety constraints with unknown dynamics. Departing from the standard optimal control framework based on dynamic programming, we study its dual formulation over the space of occupancy measures. For control-affine dynamics, with proper reparametrization, the problem can be formulated as an infinite-dimensional convex optimization over occupancy measures. Moreover, the safety constraints can be naturally captured by linear constraints in this formulation. Furthermore, this dual formulation can still be approximately obtained by utilizing the Koopman theory when the underlying dynamics are unknown. Finally, to develop a practical method to solve the resulting convex optimization, we choose a polynomial basis and then relax the problem into a semi-definite program (SDP) using sum-of-square (SOS) techniques. Simulation results are presented to demonstrate the efficacy of the developed framework.