Necessity of Vanishing Shadow Price in Infinite Horizon Control Problems

Abstract

This paper refines the necessary optimality conditions for uniformly overtaking optimal control on infinite horizon in the free end case. This condition is applicable to general non-stationary systems and the optimal objective value is not necessarily finite. In the papers of S.M. Aseev, A.V. Kryazhimskii, V.M. Veliov, K.O. Besov there was suggested a 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 version 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. Some examples are discussed.

On Hamiltonian as Limiting Gradient in Infinite Horizon Problem

“…This formula may not point towards a solution of the PMP even if the integral converges in the Lebesgue sense, see [17]. For details on the other (the more general formulas), see [16].…”

“…with the help of the control (u * , 1) . Then, y n (0) = κ( y n (τ n ), τ n ) (see ( 16 ) ). Note that y n = (x n , z n , ψ n , φ n , λ n ) satisfies ( 4a ) -( 4c ) , and ψ n (τ n ) = ψ n (τ n ) = 0 , φ n (τ n ) = φ n (τ n ) = − dhn ds ( z n (τ n ) − τ n ) .…”

“…Here, the Michel condition is used along with some limiting solution of the Pontryagin maximum principle (see [17]); the limiting solution may be considered without assumptions on the asymptotic behaviour of trajectories or adjoint variables. The idea of the limiting solution can be traced to paper [24]; see its connection with the Aseev-Kryazhimskii formula in [16].The general case of Bolza-type infinite horizon problem with free right end was studied in [17]. In this paper, we prove the existence of a limiting solution of PMP that satisfies the Michel condition for uniformly overtaking optimal control; the arising transversality conditions are expressed in the form of limiting gradients of payoff function at infinity.…”

“…Here, the Michel condition is used along with some limiting solution of the Pontryagin maximum principle (see [17]); the limiting solution may be considered without assumptions on the asymptotic behaviour of trajectories or adjoint variables. The idea of the limiting solution can be traced to paper [24]; see its connection with the Aseev-Kryazhimskii formula in [16].…”

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

“…In the papers [1,2,3,4,5], more general formula (the Aseev-Kryazhimskii formula) was proved for certain other classes of nonlinear control problems. It takes the form of an improper integral of a function, the summability of which on the whole half-line is provided by means of imposing the asymptotic conditions (similar to the dominating discount conditions) on the system (for details, refer to Subsect.3.1 of this paper or [2, Sect.16], [5], [23, Sect.6]).Another way to decrease the number of solutions of such an incomplete system of relations was proposed by Seierstad [30]. He considered a set of shortened problems and the corresponding expressions of the Maximum Principle, in each of which he obtained a co-state arc as a component of the solution of the corresponding system.…”