We consider the limit behavior, as N --> infinity, of the expression Sigma(i = 0)(N - 1) upsilon(x(i),x(i = 1)) for programs {x(i)}(infinity)(i = 0) in a compact metric space K, where upsilon is a real-valued function defined on K x K. We study the structure of (upsilon)-weakly optimal programs and establish a turnpike theorem for a generic continuous function upsilon. We also prove the existence of an almost periodic (upsilon)-optimal program for a generic continuous function upsilon.