A Deterministic Linear Quadratic Time-Inconsistent Optimal Control Problem

TL;DR

This paper introduces a method to find time-consistent equilibrium controls for linear quadratic time-inconsistent optimal control problems using a game-theoretic approach involving coupled differential equations.

Contribution

It presents a novel framework for constructing equilibrium controls in time-inconsistent problems via a family of non-cooperative differential games.

Findings

Existence of equilibrium control under certain conditions

Representation of equilibrium control through coupled differential equations

Application of game-theoretic approach to control problems

Abstract

A time-inconsistent optimal control problem is formulated and studied for a controlled linear ordinary differential equation with quadratic cost functional. A notion of equilibrium control is introduced, which can be regarded as a time-consistent solution to the original time-inconsistent problem. Under certain conditions, we constructively prove the existence of such an equilibrium control which is represented via a forward ordinary differential equation coupled with a backward Riccati--Volterra integral equation. Our constructive approach is based on the introduction of a family of $N$-person non-cooperative differential games.