Necessary Conditions In Infinite-Horizon Control Problem That Need No Asymptotic Assumptions

TL;DR

This paper derives necessary boundary conditions for infinite-horizon optimal control problems with terminal constraints that do not rely on asymptotic assumptions, broadening the applicability of optimality conditions.

Contribution

It establishes boundary conditions for co-state arcs in infinite-horizon control problems without requiring asymptotic behavior assumptions, extending existing theories.

Findings

Boundary conditions derived without asymptotic assumptions.

Conditions coincide with Aseev and Kryazhimskii's when subdifferential is singleton.

Illustrated with several practical examples.

Abstract

We consider an infinite-horizon optimal control problem with an asymptotic terminal constraint. For the the weakly overtaking criterion and the overtaking criterion, necessary boundary conditions on co-state arcs are deduced, these conditions need no assumptions about the asymptotic behavior of the motion, co-state arc, cost functional, and its derivatives. In the absence of an asymptotic terminal constraint, these boundary conditions with the Pontryagin Maximum Principle allow raising the co-state arcs, corresponding to some asymptotic subdifferentials of the cost functional (fixing the optimal control) at infinity. If this set is a singleton, these conditions coincide with the co-state arc representation proposed by Aseev and Kryazhimskii. These results are illustrated by several examples.