Stabilization of cycles with stochastic prediction-based and target-oriented control.

Area of Science:

• Dynamical Systems and Control Theory
• Stochastic Processes
• Mathematical Ecology

Background:

• Difference equations are fundamental in modeling discrete-time systems.
• Stochastic perturbations can destabilize or stabilize dynamical systems.
• Existing research on noise in control primarily focused on its destabilizing effects.

Purpose of the Study:

• To stabilize prescribed cycles or equilibria of difference equations using pulsed stochastic control.
• To demonstrate the stabilizing effect of stochastic perturbation, expanding the viable control parameter range.
• To introduce novel applications of stochastic control for prediction-based and target-oriented strategies.

Main Methods:

• Application of pulsed stochastic control inspired by Kolmogorov's law of large numbers.
• Analysis of both prediction-based and target-oriented control frameworks.
• Numerical simulations on established ecological models (logistic, Ricker, Maynard Smith).

Main Results:

• Stabilization of difference equations achieved over a wider parameter range due to noise.
• First demonstration of noise's stabilizing effect in prediction-based control.
• Introduction and successful application of a stochastic target-oriented control for point equilibria/cycles.

Conclusions:

• Pulsed stochastic control offers a robust method for stabilizing dynamical systems, particularly in ecological contexts.
• Noise, often viewed as detrimental, can be harnessed for enhanced system stability and control.
• The study expands the theoretical and practical applications of stochastic control in mathematical biology.