We prove a general existence and uniqueness result of solutions for backward stochastic differential equation (BSDE) with Lipschitz condition. also establish continuous dependence property comparison theorem to this type BSDEs, thus strengthening existing results.

Control and state estimation of nonlinear systems satisfying a Lipschitz continuity condition have been important topics in nonlinear system theory for over three decades, resulting in a substantial amount of literature. The main criticism behind this approach, however, has been the restrictive nature of the Lipschitz continuity condition and the conservativeness of the related results. This wo...

We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity is sharp ev...

An optimal control problem with quadratic cost functional for the steady-state Navier-Stokes equations with no-slip boundary condition is considered. Lipschitz stability of locally optimal controls with respect to certain perturbations of both the cost functional and the equation is proved provided a second-order sufficient optimality condition holds. For a sufficiently small Reynolds number, e...

This article presents the results on existence, uniqueness and stability of mild solutions to neutral stochastic functional evolution integro-differential equations with non-Lipschitz condition and Lipschitz condition. The existence of mild solutions for the equations are discussed by means of semigroup theory and theory of resolvent operator. Under some sufficient conditions, the results are o...

Uncertain delay differential equation (UDDE) is a type of differential equations driven by Liu process. It has been proved that uncertain delay differential equation has a unique solution in the finite domain, under the conditions that the coefficients are global Lipschitz continuous. This paper will extend this existence and uniqueness theorem from finite domain to infinite domain under the lo...

We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in R n. The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smoothability of a Lipschitz manifold in terms of a Sobolev regularity for frames ...

A recent proposal of Norton (2003) to show that a simple Newtonian system can exhibit stochastic acausal behavior by giving rise to spontaneous movements of a mass on the dome of a certain shape is examined. We discuss physical significance of an often overlooked and yet important Lipschitz condition the violation of which leads to the existence of anomalous non-trivial solutions in this and si...