New results on set stability and input-to-state stability in pulse-width modulated (PWM) control systems with disturbances are presented. The results are based on a recent generalization of two time scale stability theory to differential equations with disturbances. In particular, averaging theory for systems with disturbances is used to establish the results. The nonsmooth nature of PWM system...

We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincaré inequality. We also prove that l-valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Ni...

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where “almost everywhere” refers to the Lebesgue measure. Our main result is an extension of this theorem where the Lebesgue measure is replaced by an arbitrary measure μ. In particular we show that the differentiability properties of Lipschitz functions at μ-almost every point ar...

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak* compact and convex valued maps equipped with the Scott topology, is continuo...

We establish the sharp rate of continuity extensions ℝm-valued 1-Lipschitz maps from a subset A ℝn to on ℝn. consider several cases when there exists extension with preserved uniform distance given map. prove that if m > 1, then map is and affine only such preserving for any defined This shows striking difference case = where function has property. Another example we it possible find an same Li...

We extend the recent result of G. Godefroy which concerns existence non-norm attaining Lipschitz maps in order to characterize norm attainment toward vectors for general setting underlying space. The main theorem present paper states that on a large class metric spaces is characterized by finite-dimensionality range As an extension his counterexample, some denseness results are also shown.