Solutions to Equilibrium HJB Equations for Time-Inconsistent Deterministic Linear Quadratic Control: Characterization and Uniqueness

TL;DR

This paper characterizes and proves the uniqueness of solutions to equilibrium HJB equations in time-inconsistent deterministic linear quadratic control problems, using Riccati equations with integral terms within a game-theoretic framework.

Contribution

It introduces a novel characterization of equilibrium solutions via Riccati equations with integral terms and establishes their uniqueness for a class of time-inconsistent control problems.

Findings

Characterization of solutions using Riccati equations with integral terms

Proof of uniqueness of equilibrium solutions

Application to time-inconsistent deterministic LQ control

Abstract

In this paper we study a class of HJB equations which solve for equilibria for general time-inconsistent deterministic linear quadratic control problems within the intra-personal game theoretic framework, where the inconsistency arises from non-exponential discount functions. We characterize the solutions to the HJB equations using a class of Riccati equations with integral terms. By studying the uniqueness of solutions to the integro-differential Riccati equations, we prove the uniqueness of solutions to the equilibrium HJB equations.