Open-loop equilibrium strategy of time-inconsistent deterministic linear-quadratic problems

Wei Ji; Yuanjiao Lei; Yinju Wei; Yandan Zhang; Wei Ji; Yuanjiao Lei; Yinju Wei; Yandan Zhang

doi:10.3934/era.2026111

Electronic Research Archive

2026, Volume 34, Issue 4: 2419-2432. doi: 10.3934/era.2026111

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Research article

Open-loop equilibrium strategy of time-inconsistent deterministic linear-quadratic problems

1.
College of Science, Guiyang University, Guiyang 550005, China
2.
School of Mathematics and Big Data, Guizhou Education University, Guiyang 550018, China

Received: 19 January 2026 Revised: 09 March 2026 Accepted: 11 March 2026 Published: 23 March 2026

This paper investigates time-inconsistent linear-quadratic control problems within an open-loop framework, where the objective functional incorporates a non-exponential discounting structure. By introducing a two-point boundary value problem and a Riccati-type equation with an error function, it is established, under appropriate conditions, that the existence of an open-loop equilibrium strategy for the time-inconsistent control problem is equivalent to the existence of solutions to both the two-point boundary value problem and the Riccati-type equation. Furthermore, through an illustrative example, it is demonstrated that there exists an essential distinction between open-loop and closed-loop equilibria in time-inconsistent control problems.
- time-inconsistent,
- open-loop equilibrium,
- linear-quadratic,
- error function,
- two-point boundary value problem,
- Riccati-type equation
Citation: Wei Ji, Yuanjiao Lei, Yinju Wei, Yandan Zhang. Open-loop equilibrium strategy of time-inconsistent deterministic linear-quadratic problems[J]. Electronic Research Archive, 2026, 34(4): 2419-2432. doi: 10.3934/era.2026111

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Abstract

This paper investigates time-inconsistent linear-quadratic control problems within an open-loop framework, where the objective functional incorporates a non-exponential discounting structure. By introducing a two-point boundary value problem and a Riccati-type equation with an error function, it is established, under appropriate conditions, that the existence of an open-loop equilibrium strategy for the time-inconsistent control problem is equivalent to the existence of solutions to both the two-point boundary value problem and the Riccati-type equation. Furthermore, through an illustrative example, it is demonstrated that there exists an essential distinction between open-loop and closed-loop equilibria in time-inconsistent control problems.

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