We propose a novel recursive design scheme of state observer for lower triangular nonlinear systems. The design begins from the bottom dynamics and propagates to upper dynamics recalling the backstepping scheme for nonlinear control. The proposed class of systems is fairly general since it includes non-uniformly observable or detectable multi-output systems. The error convergence to zero is pro...

Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on Rn or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p > 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L)−1/2 on the Lp space. As an application, for 1 < p < 3+ ε, we establish the Lp boundedness...

We show that S.Vavasis’ sufficient condition for global invertibility of a polynomial mapping can be easily generalized to the case of a general Lipschitz mapping.

Influenced by Higham, Mao and Stuart [10], several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. In this ...

Lipschitz self mappings of metric spaces appear in many branches of mathematics. In this paper we introduce a modification of the Lipschitz condition which takes into account not only the mapping itself but also the behaviour of a finite number of its iterates. We refer to such mappings as mean Lipschitzian. The study of this new class of mappings seems potentially interesting and leads to some...