The reproducing kernels underlying LQ control and Kalman filtering, and their duality

Abstract

I was surprised to first show that reproducing kernels appear in linear-quadratic optimal control because of its Hilbertian vector spaces of trajectories (the cost inducing a quadratic norm), while, for estimation problems, they appear through covariances of Gaussian processes. It is actually this dual, deterministic and stochastic, nature of kernels which underlies the duality between optimal control and estimation in the Linear-Quadratic case. This viewpoint, where trajectories are more important than control functions, was first applied to tackle state-constrained LQ control problems in R^d. The approach was later extended with Alain Bensoussan (UT Dallas) to LQ parabolic PDE control, e.g. controlled heat equation, and even to mean-field LQ optimal control.