Averaged Controllability of the Random Schrödinger Equation with Diffusivity Following Absolutely Continuous Distributions

This paper is devoted to the averaged controllability of the random Schrödinger equation, with diffusivity as a random variable drawn from a general probability distribution. First, we show that the solutions to these random Schrödinger equations are null averaged controllable with an open-loop control independent of randomness from any arbitrary subset of the domain with strictly positive measure and in any time. This is done for an interesting class of random variables, including certain stable distributions, specifically recovering the known result when the random diffusivity follows a normal or Cauchy distribution. Second, by the Riemann-Lebesgue lemma, we prove for any time the lack of averaged exact controllability in a $L^2$ setting for all absolutely continuous random variables. Notably, this implies that this property is not inherited from the exact controllability of the Schrödinger equation. Third, we show that simultaneous null controllability is not possible except for a finite number of scenarios. Finally, we perform numerical simulations that robustly validate the theoretical results.

💡 Research Summary

This paper investigates the controllability of the Schrödinger equation when the diffusion coefficient is a random variable α drawn from a general probability distribution that is absolutely continuous with respect to Lebesgue measure. The state equation is

∂ₜy − α iΔy = 1_{G₀}u, y|_{∂G}=0, y(0)=y₀,

where G⊂ℝᵈ is a Lipschitz domain, G₀⊂G has positive measure, and u∈L²((0,T)×G₀) is the control. The randomness enters only through α; the control and the initial datum are deterministic. The authors focus on the average (expectation) of the solution, denoted E