Semi-Explicit Solution of Some Discrete-Time Higher-Order-Cost Mean-Field-Type Control

TL;DR

This paper develops a unified framework for solving discrete-time optimal control problems with higher-order power-law costs, providing semi-explicit solutions that account for stochasticity and mean-field effects, and revealing less aggressive control policies than quadratic costs.

Contribution

It introduces convex-completion techniques to derive semi-explicit control laws for higher-order costs in deterministic and stochastic mean-field systems, extending traditional quadratic control methods.

Findings

Higher-order costs lead to less aggressive control policies.

Variance-aware solutions are developed for systems with noise.

Numerical analyses validate the theoretical findings.

Abstract

Traditional solvable optimal control theory predominantly focuses on quadratic costs due to their analytical tractability, yet they often fail to capture critical non-linearities inherent in real-world systems including water, energy, agriculture, and financial networks. Here, we present a unified framework for solving discrete-time optimal control with higher-order state and control costs of power-law form. By building convex-completion techniques, we derive semi-explicit expressions for control laws, cost-to-go functions, and recursive coefficient dynamics across deterministic and stochastic system settings. Key contributions include variance-aware solutions under additive and multiplicative noise, extensions to mean-field-type-dependent dynamics, and conditions that ensure the positivity of recursive coefficients. In particular, we establish that higher-order costs induce less aggressive control policies compared to quadratic formulations, a finding that is validated through numerical analyses.