An Optimal Control Perspective on Externally Induced Tipping of Rigidly Shifting Systems

Abstract

The standard setting for rate-induced tipping involves fixing a particular parameterized family of smooth forcing functions and identifying a critical value of the rate parameter. In contrast, we consider a broad collection of all possible forcing functions, continuous but not necessarily smooth, and seek a general property possessed by those which effect tipping behavior. We focus on rigidly shifting asymptotically autonomous scalar systems x ̇ = f(x + λ(t)) and identify a nonsmooth choice of forcing function λ(t) which is an optimal tipping strategy in the sense that it utilizes the least possible maximum speed. Under a co-moving change of coordinates, the problem of finding this optimal λ(t) becomes dual to the problem of finding an additive control function that achieves basin escape with minimum fuel. We show the optimizer is a bang-bang control. The outcome is a lower bound on the speed |λ ̇(t)| that must be attained at least once in order to induce tipping. Its value depends on the total arclength ∞s∞|λ ̇(t)| dt of forcing, and may be interpreted as a safe threshold rate associated to each given arclength, such that if the speed of forcing remains everywhere slower than this, tipping cannot occur. The bound is tight in the sense that there exists a forcing function which induces tipping, possesses the required arclength, and never exceeds the threshold speed. Further, the threshold speed is a strictly decreasing function of arclength, thus capturing the abstract trade off between how fast and how far of a minimal disturbance characterizes tipping. While our results assume a scalar setting, the prospect of generalizing to n-dimensions is discussed and formulated as a conjecture. The control-theoretic construction used in deriving the above inspires a new theory of resilience, which is a slight modification of the intensity kof attraction framework of McGehee and Meyer. This is a family of resilience values parameterized by a number representing the allowable L1 norm of perturbations; in the limit as the integral-constraining parameter grows unbounded, these values approach the intensity of attraction. This integral-constrained intensity of attraction has the advantage of increased descriptiveness under scenarios where limited total resources are available for perturbing the system. We suggest it to be the natural choice for quantifying the resilience of a rigidly shifting system to externally-forced tipping.