On necessary boundary conditions for strictly optimal control in infinite horizon control problems

TL;DR

This paper establishes stronger boundary conditions at infinity for strictly optimal controls in infinite horizon problems, extending classic maximum principle results and providing explicit adjoint variable expressions.

Contribution

It introduces a stronger boundary condition at infinity for strict optimality, refining the maximum principle for a broad class of infinite horizon control problems.

Findings

Derived a new boundary condition at infinity for optimal controls.

Expressed the adjoint variable as an improper integral depending on the trajectory.

Extended the maximum principle to a broader class of problems.

Abstract

In the paper we consider the infinite horizon control problems on the interval with free right-hand endpoint. We obtain the necessary conditions of strict optimality. The method of the proof actually follows the classic paper by Halkin, and the boundary condition for infinity that we construct in our paper is a stronger variety of the Seierstad condition. The complete system of relations of the maximum principle that was obtained in the paper allows us to write the expression for the adjoint variable in the form of improper integral that depends only on the developing trajectory. S.M. Aseev, A.V. Kryazhimskii, and V.M. Veliov obtained the similar condition as a necessary condition for certain classes of control problems. As we note in our paper, the obtained conditions of strict optimality lead us to a redefined system of relations for sufficiently broad class of control problems. An example is considered.