  Optimal control deals with optimization problems in which variables steer a
dynamical system, and its outcome contributes to the objective function. Two
classical approaches to solving these problems are Dynamic Programming and the
Pontryagin Maximum Principle. In both approaches, Hamiltonian equations offer
an interpretation of optimality through auxiliary variables known as costates.
However, Hamiltonian equations are rarely used due to their reliance on
forward-backward algorithms across the entire temporal domain. This paper
introduces a novel neural-based approach to optimal control, with the aim of
working forward-in-time. Neural networks are employed not only for implementing
state dynamics but also for estimating costate variables. The parameters of the
latter network are determined at each time step using a newly introduced local
policy referred to as the time-reversed generalized Riccati equation. This
policy is inspired by a result discussed in the Linear Quadratic (LQ) problem,
which we conjecture stabilizes state dynamics. We support this conjecture by
discussing experimental results from a range of optimal control case studies.
