Averaged Controllability of Time-Fractional Schr\"odinger Equations with Random Quantum Diffusivity

TL;DR

This paper investigates the averaged controllability of time-fractional Schrödinger equations with random quantum diffusivity, revealing limitations for certain distributions and proposing methods to achieve controllability under specific conditions.

Contribution

It introduces a new fractional characteristic function and demonstrates null averaged controllability for a class of random variables, including the fractional biharmonic diffusion equation.

Findings

Null controllability only for a countable set of realizations.

Impossibility of simultaneous null controllability for absolutely continuous distributions.

Achieved null controllability for specific classes of random variables using an open-loop control.

Abstract

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.