Mitigation of tipping point transitions by time-delay feedback control.

Area of Science:

• Nonlinear dynamics
• Stochastic systems analysis
• Control theory

Background:

• Stochastic multistable systems are prone to transitions between equilibria due to inherent noise.
• Maintaining a system's proximity to a desired equilibrium is crucial in many applications.
• Linear delay feedback control is a potential strategy to manage these transitions.

Purpose of the Study:

• To investigate the efficacy of linear delay feedback control in mitigating transitions in stochastic multistable systems.
• To analyze the dual effects of the control term: stabilization and noise intensification.
• To derive analytical criteria for successful control parameter selection.

Main Methods:

• Analysis of stochastic differential equations under linear delay feedback.
• Derivation of analytical expressions for the control's stabilizing and destabilizing effects.
• Mathematical formulation to identify optimal control gain and delay parameters.

Main Results:

• The control term exhibits a stabilizing effect by reinforcing the potential well.
• A destabilizing effect arises from noise intensification, quantified by a factor dependent on delay and gain.
• Successful mitigation is determined by the interplay between stabilization and noise amplification.

Conclusions:

• Analytical results provide clear guidelines for selecting control gain and delay to ensure system stability.
• The derived methods eliminate the necessity for computationally intensive Monte Carlo simulations.
• The findings are validated through practical application on two distinct system examples.