Dual Effect, Certainty Equivalence, and Separation Revisited: A Counterexample and a Relaxed Characterization for Optimality

—We study the optimality of control policies admitting certainty equivalence or separation (of estimation and control) in discrete-time stochastic control, with two main contributions: (i) We ﬁrst revisit the inﬂuential theorem given in the seminal 1974 paper by Bar-Shalom and Tse which studies the equivalence between certainty equivalence (CE) and no-dual-effect (NDE) properties in discrete-time stochastic control problems involving a linear dynamic system with a possibly non-linear measurement function. We show that there is a subtle error in Bar-Shalom and Tse’s proof of the claim that CE implies NDE. Moreover, we prove that the claim does not hold by providing a counterexample. (ii) As our second and primary contribution, we introduce an alternative and a more relaxed notion of dual-freeness and establish that this new notion is sufﬁcient to guarantee the separation of estimation and control and CE in the same control problem considered by Bar-Shalom and Tse.