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arXiv · 2503.04465

Averaged Controllability of the Random Schr\"odinger Equation with Diffusivity Following Absolutely Continuous Distributions

Jon Asier Bárcena-Petisco·Fouad Et-Tahri
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Abstract

This paper is devoted to the averaged controllability of the random Schr\"odinger equation, with diffusivity as a random variable drawn from a general probability distribution. First, we show that the solutions to these random Schr\"odinger 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 Schr\"odinger 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.

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Jon Asier Bárcena-Petisco, Fouad Et-Tahri. 2025-03-06. Averaged Controllability of the Random Schr\"odinger Equation with Diffusivity Following Absolutely Continuous Distributions. https://doi.org/10.1137/25m1739698

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