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

Traditional solvable optimal control theory predominantly focuses on quadratic costs due to their analytical tractability, yet they often fail to capture critical nonlinearities 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 (lower action changes) compared to quadratic formulations, a finding that is validated through numerical analyses.