Lower closure and existence theorems for optimal control problems with infinite horizon

Lower closure theorems are proved for optimal control problems governed by ordinary differential equations for which the interval of definition may be unbounded. One theorem assumes that Cesari's property (Q) holds. Two theorems are proved which do not require property (Q), but assume either a generalized Lipschitz condition or a bound on the controls in an appropriateL_p-space. An example shows that these hypotheses can hold without property (Q) holding.