Rectification characterization of power-adapted time scales and non-autonomous discrete product quasi-semigroups

• Khadija Elgourmati

In this talk, we will present a class of non-uniform discrete time scales adapted to the power function $F(t, \alpha)=t^\alpha$, where $\alpha \in(0,1)$. We first characterize these time scales by showing that a constant ratio between the graininess and the power weight leads to the recurrence $ t_{n+1}=t_n+c t_n^\alpha . $ Using the associated conformable clock, this non-uniform time scale is transformed into the uniform grid $c \mathbb{N}_0$. This transformation makes it possible to study the problem in a simpler intrinsic time variable. We then introduce a family of non-autonomous discrete product quasi-semigroups generated by a sequence of bounded linear operators on a Hilbert space. We establish their main properties, including the deformed cocycle relation, a growth estimate, and a differentiation formula. We also obtain a discrete Duhamel formula for the corresponding non-homogeneous evolution equation. Finally, we give a condition for exact controllability in terms of the associated discrete Gramian operator. A two-dimensional example, together with a numerical simulation, is presented to illustrate the theoretical results.