We give a sufficient condition for a projective metric on a subset of a Euclidean space to admit a bi-Lipschitz embedding into Euclidean space of the same dimension.

We present a general necessary condition for separation of the reachable set of a Lipschitz control system from another given set of states, expressed in terms of an “approximating multicone” to the set in a sense that contains as special cases the Clarke and Mordukhovich cones. We then show how this separation result implies a strengthened form of the usual sufficient condition for local contr...

This paper is concerned with the stability and numerical analysis of solution to highly nonlinear stochastic differential equations with jumps. By the Itô formula, stochastic inequality and semi-martingale convergence theorem, we study the asymptotic stability in the pth moment and almost sure exponential stability of solutions under the local Lipschitz condition and nonlinear growth condition....

The Lipschitz function algebras were first defined in the 1960s by some mathematicians, including Schubert. Initially, the Lipschitz real-value and complex-value functions are defined and quantitative properties of these algebras are investigated. Over time these algebras have been studied and generalized by many mathematicians such as Cao, Zhang, Xu, Weaver, and others. Let  be a non-emp...

We propose two algorithms for solving global optimization problems on a hyperrectangle with an objective function satisfying the Vanderbei condition (this is also called $$\varepsilon $$ -Lipschitz continuous function). The belong to family of non-uniform covering methods. For we prove propositions about convergence -solution in terms function. illustrate performance using several test numerica...

In the paper, the global optimization problem of a multidimensional “black-box” function satisfying the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant is considered. A new efficient algorithm for solving this problem is presented. At each iteration of the method a number of possible Lipschitz constants is chosen from a set of values varying from zero to infinity. Th...

In this paper, we provide suitable adaptations of the “weak version of Bernstein method” introduced by the first author in 1991, in order to obtain Lipschitz regularity results and Lipschitz estimates for nonlinear integro-differential elliptic and parabolic equations set in the whole space. Our interest is to obtain such Lipschitz results to possibly degenerate equations, or to equations which...

We give, in a non-smooth setting, some conditions under which (some of) the minimizers of ∫ Ω f (∇u(x)) dx + g(x,u(x)) dx among the functions in W1,1(Ω) that lie between two Lipschitz functions are Lipschitz. We weaken the usual strict convexity assumption in showing that, if just the faces of the epigraph of a convex function f :Rn → R are bounded and the boundary datum u0 satisfies a generali...

The goal of the paper is to design sequential strategies which lead to efficient optimization of an unknown function under the only assumption that it has a finite Lipschitz constant. We first identify sufficient conditions for the consistency of generic sequential algorithms and formulate the expected minimax rate for their performance. We introduce and analyze a first algorithm called LIPO wh...

In this paper, we consider the state estimation problem for flexible joint manipulators that involve nonlinear characteristics in their stiffness. The two key ideas of our design are that (a) an accelerometer is used in order that the estimation error dynamics do not depend on nonlinearities at the link part of the manipulators and (b) the model of the nonlinear stiffness is indeed a Lipschitz ...