Boundary Observers for the Expected Value of a Randomly Switching Reaction-Diffusion PDE

The pointwise expected value of the solution of a randomly switching reaction-diffusion PDE is by itself the solution of a deterministic system of coupled reaction-diffusion equations; provided that the random switching is Markovian and that the PDE satisfies some regularity conditions. Following recent results on boundary observers for systems of coupled reaction-diffusion equations, an observer is constructed for the asymptotic estimation of the expected value of the randomly switching reaction-diffusion PDE. Although only the case where the PDE switches between two states is developed here, the same results hold for an arbitrary, but finite, number of switching states. In general, the observer gains are to be computed numerically from the solution of a system of coupled second order hyperbolic PDEs. Several phenomena described by randomly switching reaction-diffusion PDEs, for example: neurotransmitters diffusion that take into account switching between quiescent and firing states, thermostats and failure in lithium-ion batteries.