ON THE LAVRENTIEV PHENOMENON FOR OPTIMAL-CONTROL PROBLEMS WITH 2ND-ORDER DYNAMICS

The present article examines control problems in one dimension far whi ch 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 less than or equal to p < infinity) which is specified for controls to be admiss ible. Here we show that the situation is different in the case of an a utonomous running cost involving, in addition to the control, the stat e and its derivative. That is, despite the density of each space with higher exponent in those with lower exponent, the infimal cost will ge nerally depend on the choice of p if sign constraints are present.