时间分数阶薛定谔方程中随机量子扩散率的平均可控性
Averaged Controllability of Time-Fractional Schrödinger Equations with Random Quantum Diffusivity
AI总结：
本文研究了时间分数阶薛定谔方程在随机量子扩散率下的平均可控性问题，证明了绝对连续扩散率无法实现同时零可控性，并提出了一种双参数分数特征函数以实现任意时间的零平均可控性。
摘要：
本文研究了时间分数阶薛定谔方程的平均可控性问题，其中量子扩散率参数是一个具有通用概率分布的随机变量。首先，通过利用 Mittag-Leffler 函数的解析性和 Muntz 定理，我们证明系统同时零可控性只能在随机扩散率的可数集合中发生。特别地，这表明对于绝对连续的随机扩散率，同时零可控性是不可能的。接下来，我们证明对于绝对连续的随机变量，无论控制时间如何，精确平均可控性都不存在。此外，我们引入了一个新的双参数分数特征函数，使得我们可以构造一类满足任意时间零平均可控性的随机变量，无论传感器集的 Lebesgue 测度如何。这通过属于 L^∞ 的开环控制实现，且与随机参数无关。特别地，我们获得了分数双谐扩散方程的零可控性。最后，我们总结了几点值得未来研究的评注和开放问题。
英文摘要：
This paper addresses the problem of averaged controllability for the time-fractional Schrodinger equation, where the quantum diffusivity parameter is a random variable with a general probability distribution. First, by exploiting the analyticity of the Mittag-Leffler function and Muntz's theorem, we show that the simultaneous null controllability of the system can occur only for a countable set of realizations of the random diffusivity. In particular, this implies the impossibility of simultaneous null controllability for absolutely continuous random diffusivity. Next, we prove the lack of exact averaged controllability for absolutely continuous random variables, irrespective of the control time. Furthermore, we introduce a new two-parameter fractional characteristic function, which allows us to construct a class of random variables satisfying null averaged controllability at any time from any arbitrary sensor set of positive Lebesgue measure. This is achieved using an open-loop control belonging to L^\infty and independent of the random parameter. In particular, we obtain the null controllability of the fractional biharmonic diffusion equation. Finally, we conclude with several remarks and open problems that merit future investigation.
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