On a resource allocation model with infinite horizon

Abstract

In this paper we treat a resource allocation model defined on an infinite interval. We show that the solution of the corresponding problem with finite horizon cannot be extended to a solution of the infinite horizon problem, since the resource allocation problem in the unmodified setting does not have a solution on an unbounded interval. To change this situation we bring an additional state constraint into the model which contains a weight function. The new problem, called now the adapted resource allocation problem, has an optimal solution which has been identified by means of the duality concept of Klötzler.

An Existence Theorem for a Class of Infinite Horizon Optimal Control Problems

“…It should be mentioned that without posing the state constraint (115) no solution of this problem exists; cf. [26]. We verify whether the problem (112)-(116) satisfies all the conditions of the existence theorem proved before.…”

“…This model was introduced in [26] and belongs to the class of problems (P ∞ ), which becomes clear if one defines the function β as β(t) = Ce αt and notices that β ∈ L 2 (R + , e −α * t ). We refer to it as to the adapted resource allocation model.…”

Optimal vaccination strategies for an SEIR model of infectious diseases with logistic growth

“…Usually functionals of type (42) arise from infinite horizon optimal control problems which are naively approximated by truncation of the infinite horizon to a finite time interval. Lykina [11] and Lykina, Pickenhain and Wagner [12] have shown that this truncation is an improper modelling since one often cannot assure the existence of solutions. This is caused by the fact that improper spaces are chosen in which the solutions are assumed to exist.…”

“…However, the number of immunized individuals shows a larger difference: 5 282 (constant maximum vaccination rate) versus 4 781 individuals (optimal control on finite horizont). The effect with respect to the terminal behaviour of the approximate control may be unexpected on the first glance but is typical for optimal control problems which should be modelled on an infinite horizont but are truncated by a finite horizont; see [12], [11], [19], and [26]. This can be interpreted as devil-may-care solution.…”

Infinite Horizon Optimal Control Problems in the Light of Convex Analysis in Hilbert Spaces

“…It turned out that in many models it is not possible to assure the existence of an optimal solution, cf. [15] and [16]. One of the reasons is the choice of spaces in which the solution is assumed to exist, see e.g.…”