Mitigation of tipping point transitions by time-delay feedback control

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 $(1-\tau\alpha)^{-1/2}$, where $\tau$ and $\alpha$ 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.