An Existence Theorem for a Class of Infinite Horizon Optimal Control Problems

Abstract

In this paper, we deal with infinite horizon optimal control problems involving affine-linear dynamics and prove the existence of optimal solutions. The innovation of this paper lies in the special setting of the problem, precisely in the choice of weighted Sobolev and weighted Lebesgue spaces as the state and control spaces, respectively, which turns out to be meaningful for various problems. We apply the generalized Weierstraß theorem to prove the existence result. A lower semicontinuity theorem which is needed for that is shown under weakened assumptions.

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

“…Some results on the existence of optimal solutions under conditions of different kind and/or in different statements of the problem were obtained in [2,11].…”

“…By virtue of estimate (10) (note that replacing u with −u changes the trajectory x to −x, so estimate (10) also holds for the absolute value of the integral on the left-hand side), the value of the functional (in any sense) on such a pair differs from the optimal value by at most 4/(T 1 + 1). Choosing a sufficiently large T 2 (depending on T 1 ), we can make the integral analogous to (11) as large as desired. This means that condition (xi) of strong uniform integrability does not hold for Ω α for any α < α * .…”

A note on the sufficiency of the maximum principle for infinite horizon optimal control problems

“…The existence of solutions was studied by Balder, Lykina, and Pickenhain, for example. The relationship between the maximum principle and dynamic programming was investigated in the work of Sagara and Ye …”