Necessity of limiting co-state arc in Bolza-type infinite horizon problem

TL;DR

This paper derives necessary boundary conditions at infinity for optimal control in Bolza-type infinite horizon problems, using transversality conditions and Aseev–Kryazhimskii formulae, without requiring asymptotic trajectory knowledge.

Contribution

It introduces a new boundary condition at infinity and links it to limiting gradients, providing explicit co-state arc expressions for infinite horizon problems.

Findings

Derived transversality conditions at infinity for Bolza problems

Connected Aseev–Kryazhimskii formulae with limiting gradients

Provided an example illustrating explicit co-state arc expression

Abstract

We investigate necessary conditions of optimality for the Bolza-type infinite horizon problem with free right end. The optimality is understood in the sense of weakly uniformly overtaking optimal control. No previous knowledge in the asymptotic behaviour of trajectories or adjoint variables is necessary. Following Seierstad's idea, we obtain the necessary boundary condition at infinity in the form of a transversality condition for the maximum principle. Those transversality conditions may be expressed in the integral form through an Aseev--Kryazhimskii-type formulae for co-state arcs. The connection between these formulae and limiting gradients of payoff function at infinity is identified; several conditions under which it is possible to explicitly specify the co-state arc through those Aseev--Kryazhimskii-type formulae are found. For infinite horizon problem of Bolza type, an example is given to clarify the use of the Aseev--Kryazhimskii formula as explicit expression of the co-state arc.