The fractional Schrödinger equation on compact manifolds: global controllability results

Infinite memory effects on the stabilization of a Biharmonic Schrödinger equation

“…on a periodic domain T with internal control supported on an arbitrary sub-domain of T. More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solution of the associated linear system, the authors proved that system (1.7) is globally exponentially stabilizable, considering f (x, t) = −ia 2 (x)u. This property together with the local exact controllability ensures that 4NLS is globally exactly controllable on T. Lastly, the first author showed in [6] the global controllability and stabilization properties for the fractional Schrödinger equation on d-dimensional compact Riemannian manifolds without boundary (M, g),…”

“…In addition to this, recently, using another approach, the authors in [6] showed that the system (1.8) is stable, however considering a damping mechanism and some important assumptions such as the Geometric Control Condition (GCC) and Unique Continuation Property (UCP). Here, we are not able to prove that the solutions decay exponentially, however, with the approach of this article, the (GCC) and (UCP) are not required.…”

Control of fractional in-time Schrodinger equations via comprehensive Caputo derivative strategies

“…Their approach utilizes an optimization-based method that centers on minimizing appropriate functionals, and the operator A is related to the Laplacian operator. Another relevant work, presented by [4], focuses on global controllability and stabilization properties for the fractional Schrödinger equation on compact Riemannian manifolds X = M without boundary, where β = 1 and A is associated with a fractional power of the Laplace-Beltrami operator ∆ g , linked to a given metric g on M . Additionally, in the studies conducted by [11] and [12], the authors derive approximate controllability properties for fractional diffusion equations 2 with orders β belonging to the interval (0, 2), while A can be related to either classical differential operators or certain nonlocal fractional-versions of the Laplacian in bounded R N -domains.…”

“…Due to (1.4), we observe that all "frequency" multipliers in both solution representations of (1.1) can be decomposed into an oscillatory part and a monotonic decaying 'tail', which has to be controlled separately 4 . Our ultimate objective is to achieve a sufficient level of generality by allowing A to be any self-adjoint operator, while retaining fractional order equations with values of β ∈ (1, 2).…”