Infinite time optimal control and periodicity (Q1826102)

Optimal control problems: maximize \[ \limsup_{T\to \infty}(1/T)\int^{T}_{0}g(x(t),u(t))dt\quad or\quad \lim_{T\to \infty}\int^{T}_{0}e^{-\delta t}g(x(t),u(t))dt, \] associated with the smooth nonlinear system \[ \dot x(t)=X_ 0(x(t))+\sum^{r}_{i=1}u_ i(t)X_ i(x(t)),\quad t\geq 0, \] are considered. The authors show that if approximately optimal solutions have a limit point in the interior of some control set then there exist approximately optimal periodic solutions. The result is applied to the growth of linear control semigroups and to a three dimensional predator- prey harvesting model.