Existence of optimal controls for continuous time infinite horizon problems

Bell, M.L.; Sargent, R. W. H.; Vinter, R. B.

doi:10.1080/002071797223389

“…The conditions (iv) and (v) of the hypothesis are somewhat stronger than the standard convexity hypothesis guaranteeing the existence of a minimizing process that Γ(t, · ) is convex-valued for every t ∈ [0, ∞). (See Balder [6], Bates [8], Baum [9], Bell et al [11], Feinstein and Luenberger [20].) Theorem A.3.…”

Section: A Properties Of the Value Function And The Hamiltonianmentioning

confidence: 99%

Value Functions and Transversality Conditions for Infinite-Horizon Optimal Control Problems

Sagara

¹

2010

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This paper investigates the relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary and sufficient conditions for optimality with or without convexity assumptions. We first derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality that exhibit the relationship between the maximum principle and dynamic programming. We then present sufficiency theorems that are consistent with the strengthened maximum principle, employing the adjoint inequalities for the Hamiltonian and the value function. Synthesizing these results, necessary and sufficient conditions for optimality are provided for the convex case. In particular, the role of the transversality conditions at infinity is clarified.

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Value Functions and Transversality Conditions for Infinite Horizon Optimal Control Problems

Sagara

¹

2008

Communications in Computer and Information Science

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This paper investigates the relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary and sufficient conditions for optimality with or without convexity assumptions. We first derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality that exhibit the relationship between the maximum principle and dynamic programming. We then present sufficiency theorems that are consistent with the strengthened maximum principle, employing the adjoint inequalities for the Hamiltonian and the value function. Synthesizing these results, necessary and sufficient conditions for optimality are provided for the convex case. In particular, the role of the transversality conditions at infinity is clarified.

show abstract