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Proportional-Derivative (PD) control is a widely used control method in various engineering systems to enhance stability and performance. In a system with only proportional control, common issues include high maximum overshoot and oscillation, observed in both the error signal and its rate of change. This behavior can be divided into three distinct phases: initial overshoot, subsequent undershoot, and gradual stabilization.
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Stabilization of cycles with stochastic prediction-based and target-oriented control.

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This study introduces pulsed stochastic control to stabilize difference equations, leveraging noise for broader control parameter ranges. The findings apply to prediction-based and target-oriented controls, enhancing stability in ecological models.

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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.