Optimal Functional Incentives for Control: The Linear-Quadratic Case with Bilinear Incentives

TL;DR

This paper analyzes the design of fixed incentive functions for linear-quadratic control systems with bilinear incentives, providing explicit solutions and stability conditions for optimal incentives in extended horizons.

Contribution

It formalizes a bi-level optimal control framework for fixed incentives, deriving analytical solutions and stability conditions in the linear-quadratic case with bilinear incentives.

Findings

Established a necessary and sufficient stability condition for the closed-loop system.

Derived a closed-form gradient of the leader's expected cost with respect to incentives.

Provided explicit incentive characterizations in asymptotic regimes, including the infinite-horizon limit.

Abstract

We study the design of functional incentive mechanisms for dynamical systems, in which a leader designs a fixed incentive function to motivate a self-interested follower to actuate the system beneficially over an extended horizon, without real-time revision of the incentive. This stands in contrast to the adaptive paradigm, in which the incentive is itself a continuously updated control variable. We formalize the problem as a discrete-time bi-level optimal control problem and derive analytical results for the linear-quadratic case with bilinear incentives and a myopic follower. Specifically, we establish a necessary and sufficient stability condition for the induced closed-loop system, derive a closed-form expression for the gradient of the expected leader cost with respect to the incentive parameter matrix, and obtain a fully closed-form cost expression in the scalar setting. Based on the latter, explicit characterizations of the optimal incentive parameter are provided in two asymptotic regimes: the infinite-horizon limit and the limit of high follower cost. For long horizons, the optimal incentive is shown to become independent of the follower's private cost parameter, with direct implications for robust mechanism design under private information.