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

Area of Science:

• Control Theory
• Applied Mathematics
• Systems Engineering

Background:

• Traditional optimal control relies on quadratic costs, limiting applicability to complex nonlinear systems.
• Real-world systems in water, energy, agriculture, and finance exhibit critical nonlinearities.
• Existing methods struggle to capture these nonlinear dynamics effectively.

Purpose of the Study:

• To present a unified framework for discrete-time optimal control with higher-order power-law costs.
• To develop analytical solutions for control laws and cost functions in deterministic and stochastic settings.
• To address limitations of quadratic cost functions in modeling real-world nonlinearities.

Main Methods:

• Utilized convex-completion techniques to derive semi-explicit solutions.
• Developed variance-aware solutions for systems with additive and multiplicative noise.
• Extended the framework to handle mean-field-type-dependent dynamics.

Main Results:

• Derived semi-explicit expressions for control laws, cost-to-go functions, and recursive coefficients.
• Established conditions for the positivity of recursive coefficients.
• Demonstrated that higher-order costs result in less aggressive control policies (reduced action changes) than quadratic costs.

Conclusions:

• The proposed framework offers a more accurate approach to optimal control for nonlinear systems.
• Higher-order power-law costs provide a more realistic and stable control strategy.
• Findings are validated through numerical analyses, showing practical implications for various networks.