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We consider an entropy-regularized version of optimal density control ofdeterministic discrete-time linear systems. Entropy regularization, or amaximum entropy (MaxEnt) method for optimal control has attracted muchattention especially in reinforcement learning due to its many advantages suchas a natural exploration strategy. Despite the merits, high-entropy controlpolicies induced by the regularization introduce probabilistic uncertainty intosystems, which severely limits the applicability of MaxEnt optimal control tosafety-critical systems. To remedy this situation, we impose a Gaussian densityconstraint at a specified time on the MaxEnt optimal control to directlycontrol state uncertainty. Specifically, we derive the explicit form of theMaxEnt optimal density control. In addition, we also consider the case wheredensity constraints are replaced by fixed point constraints. Then, wecharacterize the associated state process as a pinned process, which is ageneralization of the Brownian bridge to linear systems. Finally, we revealthat the MaxEnt optimal density control gives the so-called Schr\"odingerbridge associated to a discrete-time linear system.

Maximum entropy optimal density control of discrete-time linear systems and Schr\"odinger bridges