Abstract
The present article examines control problems in one dimension for which there is an autonomous running cost and a specified terminal state. In this case, when the running cost involves only the control and the state, it is known that the infimal cost corresponding to any initial state is unaffected by the precise choice of $L^p$ space $(1 \leq p < \infty)$ which is specified for controls to be admissible. Here we show that the situation is different in the case of an autonomous running cost involving, in addition to the control, the state and its derivative. That is, despite the density of each space with higher exponent in those with lower exponent, the infimal cost will generally depend on the choice of $p$ if sign constraints are present.
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