Control in Probability for SDE Models of Growth Population

Area of Science:

• Stochastic analysis
• Control theory
• Optimization

Background:

• Stochastic differential equations (SDEs) model complex dynamic systems.
• Control optimization aims to find optimal strategies for system management.
• Chance-constrained problems involve probabilistic performance requirements.

Purpose of the Study:

• To develop a method for solving a chance-constrained control optimization problem for stochastic dynamics.
• To ensure a stochastic process remains bounded with high probability over a time interval.
• To adapt existing models for population growth to this control framework.

Main Methods:

• Discretization of the stochastic dynamic system.
• Restriction of control functions to piecewise mappings.
• Transformation of an infinite-dimensional problem into a finite-dimensional one.
• Analysis of well-posedness and approximation properties.

Main Results:

• The discretization and control function restriction enable the transformation of the problem.
• The transformed problem allows for analysis of well-posedness and approximation.
• Numerical simulations validate the approach using a population growth model.

Conclusions:

• The proposed discretization method provides a tractable approach to chance-constrained control optimization.
• The method is effective for managing stochastic processes within desired probability bounds.
• The framework is applicable to real-world problems, such as population dynamics.