In this paper we investigate the problem of controlling a partially observed stochastic dynamical system such that its state is difficult to infer using a (fixed-interval) Bayesian smoother. This problem arises naturally in applications in which it is desirable to keep the entire state trajectory of a system concealed. We pose our smoothing-averse control problem as the problem of maximising the (joint) entropy of smoother state estimates (i.e., the joint conditional entropy of the state trajectory given the history of measurements and controls). We show that the entropy of Bayesian smoother estimates for general nonlinear state-space models can be expressed as the sum of entropies of marginal state estimates given by Bayesian filters. This novel additive form allows us to reformulate the smoothing-averse control problem as a fully observed stochastic optimal control problem in terms of the usual concept of the information (or belief) state, and solve the resulting problem via dynamic programming. We illustrate the applicability of smoothing-averse control to privacy in cloud-based control and covert robotic navigation.
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