  This paper is concerned with optimal control problems for control systems in
continuous time, and interacting particle system methods designed to construct
approximate control solutions. Particular attention is given to the linear
quadratic (LQ) control problem. There is a growing interest in re-visiting this
classical problem, in part due to the successes of reinforcement learning (RL).
The main question of this body of research (and also of our paper) is to
approximate the optimal control law {\em without} explicitly solving the
Riccati equation. A novel simulation-based algorithm, namely a dual ensemble
Kalman filter (EnKF), is introduced. The algorithm is used to obtain formulae
for optimal control, expressed entirely in terms of the EnKF particles. An
extension to the nonlinear case is also presented. The theoretical results and
algorithms are illustrated with numerical experiments.
