Necessity of vanishing shadow price in infinite horizon control problems

TL;DR

This paper establishes the necessity of a vanishing shadow price boundary condition for infinite horizon optimal control problems with free end, extending previous results to more general non-stationary systems.

Contribution

It proves a more general geometric boundary condition necessary for optimality, applicable to non-stationary systems with possibly infinite objectives.

Findings

The boundary condition is necessary for uniformly overtaking optimal control.

Under certain assumptions, the boundary condition ensures a unique Lagrange multiplier.

The results apply to general non-stationary systems with possibly infinite objectives.

Abstract

This paper investigates the necessary optimality conditions for uniformly overtaking optimal control on infinite horizon in the free end case. %with free right endpoint. In the papers of S.M.Aseev, A.V.Kryazhimskii, V.M.Veliov, K.O.Besov there was suggested the boundary condition for equations of the Pontryagin Maximum Principle. Each optimal process corresponds to a unique solution satisfying the boundary condition. Following A.Seierstad's idea, in this paper we prove a more general geometric variety of that boundary condition. We show that this condition is necessary for uniformly overtaking optimal control on infinite horizon in the free end case. A number of assumptions under which this condition selects a unique Lagrange multiplier is obtained. The results are applicable to general non-stationary systems and the optimal objective value is not necessarily finite. Some examples are discussed.