[Paper Review] Necessary Conditions for Infinite Horizon Optimal Control Problems Revisited

Anton O. Belyakov|arXiv (Cornell University)|Dec 2, 2015

Economic theories and models22 citations

TL;DR

This paper revisits necessary optimality conditions for infinite-horizon optimal control problems, particularly when the objective functional may diverge. It introduces a new transversality condition that avoids explicit dependence on the adjoint variable and proves it necessary under overtaking and weakly overtaking optimality, outperforming classical conditions that fail in cases like the Ramsey problem without discounting.

ABSTRACT

Necessary optimality conditions in the form of the maximum principle for control problems with infinite time horizon are considered. Both finite and infinite values of objective functional are allowed since the concept of overtaking or weakly overtaking optimality is used. New form of optimality condition is obtained and compared with the transversality conditions usually used in the literature. The examples, where these transversality conditions may fail while the new condition holds are presented. For Ramsey problem of capital accumulation a simple form of necessary optimality conditions is derived, which is also valid in the case of zero discounting.

Motivation & Objective

• To address the failure of classical transversality conditions in infinite-horizon optimal control problems where the objective functional may diverge.
• To develop a new necessary optimality condition that remains valid even when standard conditions like lim ψ(t) = 0 or lim H = 0 fail.
• To provide a robust framework for necessary conditions in problems with overtaking and weakly overtaking optimality, especially in economic growth models.
• To extend the applicability of the maximum principle to cases such as the Ramsey problem under zero discounting, where traditional conditions are insufficient.
• To derive a condition independent of the adjoint variable, improving generality and avoiding reliance on restrictive assumptions like interior optimal trajectories.

Proposed method

• Introduces overtaking and weakly overtaking optimality as generalized criteria for infinite-horizon problems, allowing for divergent or oscillating objective functionals.
• Derives a new necessary condition using a variational approach with needle variations, avoiding explicit dependence on the adjoint variable.
• Employs a modified Hamiltonian structure and a Cauchy-type formula for the adjoint variable, ensuring consistency under non-convergent integrals.
• Uses the fundamental matrix of the linearized state equation to express the adjoint variable and derive the new transversality condition.
• Applies the new condition to the Ramsey problem with zero discounting, deriving a simple, valid necessary condition where classical ones fail.
• Proves the new condition subsumes classical forms (e.g., condition (5)) as special cases, extending their domain of applicability.

Experimental results

Research questions

• RQ1 Why do classical transversality conditions such as lim ψ(t) = 0 or lim H = 0 fail in infinite-horizon problems with divergent objective functionals?
• RQ2 Can a necessary optimality condition be formulated that remains valid when the objective functional does not converge?
• RQ3 How can the maximum principle be extended to cases like the Ramsey problem under zero discounting, where standard conditions are inapplicable?
• RQ4 Is it possible to derive a necessary condition that does not explicitly depend on the adjoint variable while still being valid under overtaking optimality?
• RQ5 In what sense does the new condition generalize or subsume existing transversality conditions like those in (5) or (4)?

Key findings

• The new transversality condition is necessary for overtaking and weakly overtaking optimality, even when the objective functional diverges.
• Classical conditions such as lim ψ(t) = 0, lim H = 0, and limsup ⟨ψ(t), x(t)⟩ ≥ liminf ⟨ψ(t), x̂(t)⟩ fail in the Ramsey problem without discounting, while the new condition holds.
• The new condition is derived without explicit dependence on the adjoint variable, making it more robust in non-interior solution cases.
• For the Ramsey problem with zero discounting, the paper derives a simple, valid necessary condition that is not captured by standard transversality forms.
• The new condition generalizes condition (5) from [1–4], extending its validity beyond problems with interior optimal trajectories.
• The proof shows that the new condition is consistent with the maximum principle and subsumes existing forms as special cases, enhancing theoretical robustness.