Fundamental performance limitations with Kullback-Leibler control cost

This research concerns fundamental performance limitations in control of discrete time nonlinear systems. The fundamental limitations are expressed in terms of the average cost of an infinite horizon optimal control problem. The control cost is defined by using a certain Kullback-Leibler divergence metric recently introduced by Todorov. The limitations are obtained via analysis of a linear eigenvalue problem defined only by the open loop dynamics. For a linear time invariant (LTI) system the fundamental limitation is shown to depend upon the unstable eigenvalues, as in the classical Bode formula. For a more general class of nonlinear systems, it is shown that the limitation arise only if the open-loop dynamics are non-ergodic.