  This paper introduces a framework for solving time-autonomous nonlinear
infinite horizon optimal control problems, under the assumption that all
minimizers satisfy Pontryagin's necessary optimality conditions. In detail, we
use methods from the field of symplectic geometry to analyze the eigenvalues of
a Koopman operator that lifts Pontryagin's differential equation into a
suitably defined infinite dimensional symplectic space. This has the advantage
that methods from the field of spectral analysis can be used to characterize
globally optimal control laws. A numerical method for constructing optimal
feedback laws for nonlinear systems is then obtained by computing the
eigenvalues and eigenvectors of a matrix that is obtained by projecting the
Pontryagin-Koopman operator onto a finite dimensional space. We illustrate the
effectiveness of this approach by computing accurate approximations of the
optimal nonlinear feedback law for a Van der Pol control system, which cannot
be stabilized by a linear control law.
