Existence of Overtaking Optimal Trajectories for Problems with Convex Integrands

In this work we study the existence of optimal overtaking trajectories for autonomous control systems on an infinite time interval. We extend, in two directions, the research initiated by Brock and Haurie (Brock, W. A., A. Haurie. 1976. On existence of overtaking optimal trajectories over an infinite time horizon. Math. Oper. Res. 1 337--346.). First we relax the technical assumptions needed to establish the existence of overtaking optimal trajectories. The assumptions in our existence result are basically those made in (Brock, W. A., A. Haurie. 1976. On existence of overtaking optimal trajectories over an infinite time horizon. Math. Oper. Res. 1 337--346.) for the existence of weakly overtaking optimal trajectories, with the additional assumption that a certain linear differential equation determined by the integrand has no nontrivial periodic solutions. A discussion and an example concerning this assumption are provided. In particular we show that this property is generic in the epi-convergence topology. Secondly, we consider convex integrands which grow uniformly to infinity rather than integrands with compact domains.