# On Balder's Existence Theorem for Infinite-Horizon Optimal Control Problems

## TL;DR

This paper extends Balder's 1983 existence theorem for infinite-horizon optimal control problems to improper integrals, relaxing uniform integrability conditions and providing a simpler proof approach with an illustrative example.

## Contribution

It generalizes Balder's theorem by weakening integrability conditions and offers a new, simpler proof method for the existence result.

## Key findings

- Extended existence theorem to improper integrals
- Relaxed uniform integrability conditions
- Provided an illustrative example

## Abstract

Balder's well-known existence theorem (1983) for infinite-horizon optimal control problems is extended to the case when the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part $\max\{f_0,0\}$ of the utility function (integrand) $f_0$ is relaxed to the requirement that the integrals of $f_0$ over intervals $[T,T']$ be uniformly bounded from above by a function $\omega(T,T')$ such that $\omega(T,T')\to 0$ as $T,T'\to\infty$. This requirement was proposed by A.V. Dmitruk and N.V. Kuz'kina (2005); however, the proof in the present paper does not follow their scheme but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.