Mitigation of tipping point transitions by time-delay feedback control

Abstract

In stochastic multistable systems driven by the gradient of a potential, transitions between equilibria is possible because of noise. We study the ability of linear delay feedback control to mitigate these transitions, ensuring that the system stays near a desirable equilibrium. For small delays, we show that the control term has two effects: i) a stabilizing effect by deepening the potential well around the desirable equilibrium, and ii) a destabilizing effect by intensifying the noise by a factor of (1 − τ α) −1/2 , where τ and α denote the delay and the control gain, respectively. As a result, successful mitigation depends on the competition between these two factors. We also derive analytical results that elucidate the choice of the appropriate control gain and delay that ensure successful mitigations. These results eliminate the need for any Monte Carlo simulations of the stochastic differential equations, and therefore significantly reduce the computational cost of determining the suitable control parameters. We demonstrate the application of our results on two examples.Many complex systems, such as the climate, stock markets, population of species, and the nervous system, have multiple stable equilibrium states. Near the tipping point of the system, small amounts of stochastic disturbances can lead to transitions between these equilibria. Typically, one of the equilibria is desirable and transitions away from it lead to catastrophic consequences. Here, we analyze the ability of a delay feedback control to mitigate transitions away from a desirable equilibrium. arXiv:1911.05292v2 [math.OC]

Investigating climate tipping points under various emission reduction and carbon capture scenarios with a stochastic climate model

“…Extreme events, such as tipping point transitions, have been the subject of much research, with an emphasis on their causal mechanisms [39][40][41], probabilistic quantification [42][43][44][45], and data-driven prediction [18,[46][47][48]. Only recently, control strategies for mitigating extreme events have been proposed [34,46,49]. In particular, Farazmand [34] proposes a time-delay feedback control for mitigating noise-induced transitions in multistable systems.…”

“…For the CO 2 sinks, we allow for a time delay τ to model the time between carbon capture and its effect being felt in the atmospheric concentration. This type of delay is typical in control theory, where there is often a lag between a modifying action and its actualization [33,34]. If this delay is non-existent or negligible, one can set τ = 0.…”

“…Only recently, control strategies for mitigating extreme events have been proposed [34,46,49]. In particular, Farazmand [34] proposes a time-delay feedback control for mitigating noise-induced transitions in multistable systems. This control strategy relies on the stationary equilibrium density of the system.…”

“…Given the time-dependent emission rate β(t) and the linearity of the CO 2 model (2.6), it does not possess such an equilibrium density. Consequently, the framework of [34] is not applicable here.…”

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