  This paper is concerned with data-driven optimal control of nonlinear
systems. We present a convex formulation to the optimal control problem (OCP)
with a discounted cost function. We consider OCP with both positive and
negative discount factor. The convex approach relies on lifting nonlinear
system dynamics in the space of densities using the linear Perron-Frobenius
(P-F) operator. This lifting leads to an infinite-dimensional convex
optimization formulation of the optimal control problem. The data-driven
approximation of the optimization problem relies on the approximation of the
Koopman operator using the polynomial basis function. We write the approximate
finite-dimensional optimization problem as a polynomial optimization which is
then solved efficiently using a sum-of-squares-based optimization framework.
Simulation results are presented to demonstrate the efficacy of the developed
data-driven optimal control framework.
