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

Balder's well-known existence theorem (1983) for infinite-horizon optimal control problems is extended to the case in which 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 of the utility function (integrand in the integral to be maximized) is relaxed to a uniform one-sided upper estimate for the integrals of the utility function over finite intervals by a quantity that depends only on the interval of integration and tends to zero as both endpoints of the interval tend to infinity. The latter condition and the idea of using it in an existence theorem for infinite-horizon optimal control problems were proposed (in the case of systems affine in control) by A.V. Dmitruk and N.V. Kuz'kina (2005).