随机薛定谔方程中扩散系数服从绝对连续分布的平均可控性
Averaged Controllability of the Random Schrödinger Equation with Diffusivity Following Absolutely Continuous Distributions
AI总结：
本文研究了随机薛定谔方程在扩散系数服从绝对连续分布时的平均可控性，证明了在特定条件下零可控性成立，并指出在$ L^2 $设定下不存在平均精确可控性，同时通过数值模拟验证了理论结论。
摘要：
本文致力于研究随机薛定谔方程的平均可控性，其中扩散系数作为从一般概率分布中抽取的随机变量。首先，我们证明这些随机薛定谔方程的解在任意子域中严格正测度的任意子集上，可以使用与随机性无关的开环控制实现零平均可控性，这适用于一类有趣的随机变量，包括某些稳定分布，特别是当随机扩散系数服从正态或柯西分布时，恢复已知结果。其次，通过黎曼-勒贝格引理，我们证明在$ L^2 $设定下，对于所有绝对连续随机变量，在任意时间均不存在平均精确可控性。值得注意的是，这表明该性质并非继承自薛定谔方程的精确可控性。第三，我们证明除了有限数量的场景外，同时零可控性是不可能实现的。最后，我们进行了数值模拟，结果稳健地验证了理论结果。
英文摘要：
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.
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