Undiscounted infinite horizon optimization problems are intrinsically difficult because (i) the objective functional may not converge; (ii) boundary conditions at the infinite ter- minal time cannot be rigorously expressed in the real number field. In this paper, by ex- tending real numbers to hyper-real numbers, we derive the optimal solution to an undis- counted infinite horizon optimization problem that has an infinite objective functional. We demonstrate that under a hyper-real terminal time, there exists a unique optimal so- lution in the hyper-real number field. We show that under fairly general conditions, the standard part of the hyper-real optimal path is the optimum among all feasible paths in the standard real number field, in the sense of two modified overtaking criteria. We also examine the applicability of our approach by considering two parametric examples.
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