Choose $q\in ℂ$ with $0\<\left|q\right|\<1$. The main theme of this paper is the study of linear $q$-difference equations over the field $K$ of germs of meromorphic functions at $0$. A systematic treatment of classification and moduli is developed. It turns out that a difference module $M$ over $K$ induces in a functorial way a vector bundle $v\left(M\right)$ on the Tate curve $E_q:={ℂ}^{*}/q^{\mathbb{Z}}$ that was known for modules with ”integer slopes“, [Saul, 2]). As a corollary one rediscovers Atiyah’s classification $\left(\right[\mathrm{A}t\left]\right)$ of the indecomposable vector bundles on the complex Tate...

In this paper, we investigate the problem of global output-feedback regulation for a class of switched nonlinear systems with unknown linear growth condition and uncertain output function. Based on the backstepping method, an adaptive output-feedback controller is designed to guarantee that the state of the switched nonlinear system can be globally regulated to the origin while maintaining global boundedness of the resulting closed-loop switched system under arbitrary switchings. A numerical example...

In this paper, the problem of global finite-time stabilization via output-feedback is investigated for a class of stochastic nonlinear cascaded systems (SNCSs). First, based on the adding a power integrator technique and the homogeneous domination approach, a global output-feedback finite-time control law is constructed for the driving subsystem. Then, based on homogeneous systems theory, it is shown that under some mild conditions the global finite- time stability in probability of the driving...