Blowup from randomly switching between stable boundary conditions for the heat equation

We find a pair of boundary conditions for the heat equation such that the solution goes to zero for either boundary condition, but if the boundary condition randomly switches, then the average solution grows exponentially in time. Specifically, we prove that the mean of the random solution grows exponentially under certain mild assumptions, and we use formal asymptotic methods to argue that the random solution grows exponentially almost surely. To our knowledge, this could be the first PDE example showing that randomly switching between two globally asymptotically stable systems can produce a blowup. We devise several methods to analyze this random PDE. First, we use the method of lines to approximate the switching PDE by a large number of switching ODEs and then apply recent results to determine if they grow or decay in the limit of fast switching. We then use perturbation theory to obtain more detailed information on the switching PDE in this fast switching limit. To understand the case of finite switching rates, we characterize the parameter regimes in which the first and second moments of the random PDE grow or decay. This moment analysis reveals rich dynamical behavior, including a region of parameter space in which the mean of the random PDE oscillates with ever increasing amplitude for slow switching rates, grows exponentially for fast switching rates, but decays to zero for intermediate switching rates. We also highlight cases in which the second moment is necessary to understand the switching system’s qualitative behavior, rather than just the mean. Finally, we give a PDE example in which randomly switching between two unstable systems produces a stable system. All of our analysis is accompanied by numerical simulation.

Keywords

piecewise deterministic Markov process, switched dynamical systems, stochastic hybrid system, random PDE, thermostat model

2010 Mathematics Subject Classification

35B44, 35K05, 35R60, 60H15, 93E15

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The author was supported by NSF grant DMS-RTG 1148230.

Received 18 August 2017

Accepted 3 April 2018

Published 31 October 2018