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Optimal control is an essential tool for stabilizing complex nonlinearsystems. However, despite the extensive impacts of methods such as recedinghorizon control, dynamic programming and reinforcement learning, the design ofcost functions for a particular system often remains a heuristic-driven processof trial and error. In this paper we seek to gain insights into how the choiceof cost function interacts with the underlying structure of the control systemand impacts the amount of computation required to obtain a stabilizingcontroller.  We treat the cost design problem as a two-step process where the designerspecifies outputs for the system that are to be penalized and then modulatesthe relative weighting of the inputs and the outputs in the cost. Tocharacterize the computational burden associated to obtaining a stabilizingcontroller with a particular cost, we bound the prediction horizon required byreceding horizon methods and the number of iterations required by dynamicprogramming methods to meet this requirement. Our theoretical results highlighta qualitative separation between what is possible, from a design perspective,when the chosen outputs induce either minimum-phase or non-minimum-phasebehavior. Simulation studies indicate that this separation also holds formodern reinforcement learning methods.

On the Computational Consequences of Cost Function Design in Nonlinear Optimal Control