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

Julian Barreiro-Gomez, Tyrone E. Duncan, Bozenna Pasik-Duncan, Hamidou Tembine

TL;DR

This paper addresses discrete-time mean-field-type optimal control with higher-order, power-law costs, moving beyond traditional quadratic formulations. It introduces a convex-completion-based framework that yields semi-explicit control laws, cost-to-go coefficients, and backward recursions for a broad class of even-order costs $x^{2p}$ and $u^{2p}$ under deterministic, additive-noise, multiplicative-noise, and mean-field settings. Key contributions include variance-aware extensions, multiplicative-noise handling, and a verification method in the space of measures, with rigorous positivity conditions ensuring well-posed solutions. The approach reveals that higher-order costs tend to produce less aggressive controls than quadratic costs, and numerical experiments illustrate the framework's applicability to water, energy, agriculture, and financial-network contexts. Overall, the work broadens solvable mean-field-type control problems to non-linear, risk-sensitive objectives with semi-explicit, tractable solutions.

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.

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

TL;DR

This paper addresses discrete-time mean-field-type optimal control with higher-order, power-law costs, moving beyond traditional quadratic formulations. It introduces a convex-completion-based framework that yields semi-explicit control laws, cost-to-go coefficients, and backward recursions for a broad class of even-order costs and under deterministic, additive-noise, multiplicative-noise, and mean-field settings. Key contributions include variance-aware extensions, multiplicative-noise handling, and a verification method in the space of measures, with rigorous positivity conditions ensuring well-posed solutions. The approach reveals that higher-order costs tend to produce less aggressive controls than quadratic costs, and numerical experiments illustrate the framework's applicability to water, energy, agriculture, and financial-network contexts. Overall, the work broadens solvable mean-field-type control problems to non-linear, risk-sensitive objectives with semi-explicit, tractable solutions.

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.
Paper Structure (17 sections, 8 theorems, 101 equations, 5 figures, 2 tables)

This paper contains 17 sections, 8 theorems, 101 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. How does the Convex Completion Works?
  3. Convexity of higher-order terms
  4. Structure of the paper
  5. Deterministic control with higher-order costs
  6. Optimal Control with Quartic Cost
  7. Extension to $\bar{x}^{2p}$ Costs
  8. Variance-aware control with additive noise
  9. Variance-aware Quartic Cost with additive noise
  10. Extension to $\bar{x}^{2p}$ Costs
  11. Variance-Aware Stochastic Cost with Multiplicative Noise
  12. Extension to $\bar{x}^{2p}$ Costs
  13. Higher-Order Moment Costs with Multiplicative State-Control and Mean-Field Type Dependent Noise
  14. Verification Method in the Space of Measures
  15. Proofs
  16. ...and 2 more sections

Key Result

Lemma 1

Let $p\geq 1, a\neq 0, b\neq 0.$ Then the mapping $z \mapsto f(z) = z^{2p} + (az+b)^{2p}$ is strictly convex. $\square$

Figures (5)

  • Figure 1: Results example corresponding to Proposition \ref{['propos:problem1']} and Proposition \ref{['propos:problem2']}.
  • Figure 2: Results example corresponding to Proposition \ref{['propos:proposition_3']} and Proposition \ref{['propos:proposition_4']}.
  • Figure 3: Results example corresponding to Proposition \ref{['propos:proposition_5']} and Proposition \ref{['propos:proposition_6']}. The noise $\epsilon$ for this example is the same one presented in Figure \ref{['fig:p3_4']}.
  • Figure 4: Results example corresponding to Proposition \ref{['propos:proposition_7']}. The noise $\epsilon$ for this example is the same one presented in Figure \ref{['fig:p3_4']}.
  • Figure 5: Comparison results: Case 1: simple controller with finite-set control actions, Case 2: state-feedback controller, and Case 3: risk-aware controller.

Theorems & Definitions (22)

  • Lemma 1
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Remark 2
  • Proposition 3
  • Remark 3
  • Proposition 4
  • Remark 4
  • Proposition 5
  • ...and 12 more