Uniform Lipschitz Research Articles

Remarques sur le calcul symbolique dans certains espaces de Besov à valeurs vectorielles

We are interested in the superposition operators T f (g):=f∘g on vector valued Besov and Lizorkin-Triebel spaces of positive smoothness exponent s. As a first step towards the characterization of functions which operate, we establish that the local Lipschitz continuity of f is necessary if the space B p,q s (ℝ n ,ℝ m ) or F p,q s (ℝ n ,ℝ m ) is imbedded into L ∞ (ℝ n ,ℝ m ), and that the uniform Lipschitz continuity of f is necessary if the space is not imbedded into L ∞ (ℝ n ,ℝ m ). We prove also that the local membership to the same space is necessary for m≤n. We finally study the regularity of the superposition operator T f .

Adaptive coupled synchronization of non-autonomous systems in ring networks

The adaptive coupled synchronization method for non-autonomous systems is proposed. This method can avoid estimating the value of coupling coefficient. Under the uniform Lipschitz assumption, we derive the asymptotical synchronization for a general coupling ring network with N identical non-autonomous systems, even when N is large enough. Strict theoretical proofs are given. Numerical simulations illustrate the effectiveness of the present method.

Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries

Here we study the asymptotic limits of solutions of some singularly perturbed elliptic systems. The limiting problems involve multiple valued harmonic functions or, in general, harmonic maps to singular spaces and free interfaces between supports of various components of the maps. The main results of the paper are the uniform Lipschitz regularity of solutions as well as the regularity of free interfaces.

Adaptive chaos control and synchronization in only locally Lipschitz systems

In the existing results on chaos control and synchronization based on the adaptive controlling technique (ACT), a uniform Lipschitz condition on a given dynamical system is always assumed in advance. However, without this uniform Lipschitz condition, the ACT might be failed in both theoretical analysis and in numerical experiment. This Letter shows how to utilize the ACT to get a rigorous control for the system which is not uniformly Lipschitz but only locally Lipschitz, and even for the system which has unbounded trajectories. In fact, the ACT is proved to possess some limitation, which is actually induced by the nonlinear degree of the original system. Consequently, a piecewise ACT is proposed so as to improve the performance of the existing techniques.

Uniqueness of solutions to weak parabolic equations for measures

We study uniqueness of solutions of parabolic equations for measures μ(dt dx) = μt(dx)dt of the type L* μ = 0, satisfying μt → ν as t → 0, where each μt is a probability measure on R d , L = ∂ t + a i j ( t , x ) ∂ x i ∂ x j + b i ( t , x ) ∂ x j is a differential operator on (0, T) × ℝd and ν is a given initial measure. One main result is that uniqueness holds under uniform ellipticity and Lipschitz conditions on aij but for bi merely local integrability and coercivity conditions are sufficient.

Lp Solutions of BSDEs with Stochastic Lipschitz Condition

We are concerned with the solutions of a special class of backward stochastic differential equations which are driven by a Brownian motion, where the uniform Lipschitz continuity is replaced by a stochastic one. We prove the existence and uniqueness of the solution in Lp with p>1.

The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay

This paper is devoted to build the existence-and-uniqueness theorem of solutions to stochastic functional differential equations with infinite delay (short for ISFDEs) at phase space BC ( ( − ∞ , 0 ] ; R d ) . Under the uniform Lipschitz condition, the linear growth condition is weaked to obtain the moment estimate of the solution for ISFDEs. Furthermore, the existence-and-uniqueness theorem of the solution for ISFDEs is derived, and the estimate for the error between approximate solution and accurate solution is given. On the other hand, under the linear growth condition, the uniform Lipschitz condition is replaced by the local Lipschitz condition, the existence-and-uniqueness theorem is also valid for ISFDEs on [ t 0 , T ] . Moreover, the existence-and-uniqueness theorem still holds on interval [ t 0 , ∞ ) , where t 0 ∈ R is an arbitrary real number.

Dual representation as stochastic differential games of backward stochastic differential equations and dynamic evaluations

In this Note, assuming that the generator is uniform Lipschitz in the unknown variables, we relate the solution of a one dimensional backward stochastic differential equation with the value process of a stochastic differential game. Under a domination condition, an F -consistent evaluation is also related to a stochastic differential game. This relation comes out of a min–max representation for uniform Lipschitz functions in terms of affine functions. The extension to reflected backward stochastic differential equations is also included. To cite this article: S. Tang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

A note on non-existence results for semi-linear cooperative elliptic systems via moving spheres

In this note, we extend some earlier non-existence, monotonicity and one-dimensionality results of W. Reichel and the author, by replacing the local Lipschitz continuity hypothesis on the non-linearities by a so-called boundedly uniform Lipschitz condition in the magnitude of u \mathbf {u} .

A nested Matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems

A new infinitesimal sufficient condition is given for uniform global asymptotic stability (UGAS) for time-varying nonlinear systems. It is used to show that a certain relaxed persistency of excitation condition, called uniform /spl delta/-persistency of excitation (U/spl delta/-PE), is sufficient for uniform global asymptotic stability in certain situations. U/spl delta/-PE of the right-hand side of a time-varying differential equation is also shown to be necessary under a uniform Lipschitz condition. The infinitesimal sufficient condition for UGAS involves the inner products of the flow field with the gradients of a finite number of possibly sign-indefinite, locally Lipschitz Lyapunov-like functions. These inner products are supposed to be bounded by functions that have a certain nested, or triangular, negative semidefinite structure. This idea is reminiscent of a previous idea of Matrosov who supplemented a Lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is definitely nonzero where the derivative of the Lyapunov function is zero. For this reason, we call the main result a nested Matrosov theorem. The utility of our results on stability analysis is illustrated through the well-known case-study of the nonholonomic integrator.

Discontinuous Galerkin methods for coupled flow and reactive transport problems

Primal discontinuous Galerkin methods with interior penalty are proposed to solve the coupled system of flow and reactive transport in porous media, which arises from many applications including miscible displacement and acid-stimulated flow. A cut-off operator is introduced in the discontinuous Galerkin schemes to treat the coupling of flow and transport and the coupling of transport and reaction. The uniform positive definitiveness and the uniform Lipschitz continuity are established for the commonly used dispersion–diffusion tensor. Interestingly, the polynomial degrees of approximation for the flow and the transport equations need to be in the same order in order to maintain the convergence of DG applied to the coupled system. Optimal or nearly optimal convergences for both flow and transport are obtained when the same polynomial degrees of approximation are chosen for flow and transport. That is, error estimate in L 2 ( H 1 ) for concentration is optimal in h and nearly optimal in p with a loss of 1 2 ; error estimates in semi- L ∞ ( H 1 ) for pressure and in L ∞ ( L 2 ) for velocity establish optimality in h and sub-optimality in p by 1 2 ; error estimates for concentration jump and pressure jump are optimal in both h and p.

Stability criteria of impulsive differential systems

This paper discusses the uniform eventual Lipschitz stability and uniform eventual asymptotic stability of impulsive differential system. Some criteria of such stability for impulsive differential system are given. Compared with the existing result, the condition is less conservative. Our technique depends on Lyapunov's direct method and comparison principle.

A construction of a Lipschitz continuous minimizer of a free boundary problem

The free boundary problem of the variational type studied by Alt and Caffarelli is investigated here. Especially, treated is the quadratic functional having the coefficient of diagonal form and the characteristic function term. The main result is to construct a minimizer with the Lipschitz continuity, which will play an essential role for proving the regularity of the free boundary. This is established by treating the regularized functional which approximates the original functional in the sense of Γ-convergence. The most important part of this paper is to show the uniform Lipschitz continuity of the minimizers of the regularized functionals. Taking advantage of the minimality, not resorting to the Euler–Lagrange equation, the minimizers turn out to belong to DeGiorgi class uniformly, the uniform Hölder estimate is obtained, and, as a result, the Lipschitz estimate is attained with the aid of the uniform Harnack inequality.

A singular perturbation problem for the p-Laplace operator

In this paper we initiate the study of the nonlinear one phase singular perturbation problem divNjru j p 2 ru OENu O; N1<p<1O in a domain ⁄ of R N. We prove uniform Lipschitz regularity of uniformly bounded solutions. Once this is done we can pass to the limit to obtain a solution to the stationary case of a combus- tion problem with a nonlinearity of power type. (The casepE 2 has been considered earlier by several authors.)

On Natural Functions and Lipschitz Functions

We give certain characterizations of the so-called natural functions introduced by Chistyakov. We show that the set of natural functions from $[a, b]$ into a normed space $X$ is small in the porosity sense in the set of uniform Lipschitz 1 functions with the norm inherited from the space of $BV$ functions. The sizes of other classes of Lipschitz functions in $BV$ are also established.

A pathological example of a uniform quotient mapping between Euclidean spaces

A uniform quotient Lipschitz mapping between Euclidean spaces of dimensionsn andn−1, which annihilates the unit ball of a hyperplane, is constructucted.

Limit theorems for iterated random functions by regenerative methods

Let ( X,d) be a complete separable metric space and ( F n ) n⩾0 a sequence of i.i.d. random functions from X to X which are uniform Lipschitz, that is, L n= sup x≠y d(F n(x),F n(y))/d(x,y)<∞ a.s. Providing the mean contraction assumption E log + L 1<0 and E log + d(F 1(x 0),x 0)<∞ for some x 0∈ X , it was proved by Elton (Stochast. Proc. Appl. 34 (1990) 39–47) that the forward iterations M n x = F n ∘⋯∘ F 1( x), n⩾0, converge weakly to a unique stationary distribution π for each x∈ X . The associated backward iterations M ̂ n x=F 1∘⋯∘F n(x) are a.s. convergent to a random variable M ̂ ∞ which does not depend on x and has distribution π. Based on the inequality d( M ̂ n+m x, M ̂ n x)⩽ exp(∑ k=1 n log L k)d(F n+1∘⋯∘F n+m(x),x) for all n, m⩾0 and the observation that ( ∑ k=1 n log L k) n⩾0 forms an ordinary random walk with negative drift, we will provide new estimates for d( M ̂ ∞, M ̂ n x) and d( M n x , M n y ), x,y∈ X , under polynomial as well as exponential moment conditions on log(1+ L 1) and log(1+ d( F 1( x 0), x 0)). It will particularly be shown, that the decrease of the Prokhorov distance between P n ( x,·) and π to 0 is of polynomial, respectively exponential rate under these conditions where P n denotes the n-step transition kernel of the Markov chain of forward iterations. The exponential rate was recently proved by Diaconis and Freedman (SIAM Rev. 41 (1999) 45–76) using different methods.

Qualitative properties of nonlinear difference systems

This paper gives criteria for the existence and uniqueness of solutions to two and multipoint boundary value problems associated with a typical system of kth order nonlinear Lyapunov system. A variation of parameters formula for nonlinear kth order system is established. Using this formula, we also study various concepts of stability and establish sufficient conditions for the kth order difference system to be Lipschitz stable and uniform Lipschitz stable.

A theorem for UGAS and ULES of (passive) nonautonomous systems: robust control of mechanical systems and ships

AbstractThe main contribution of this paper is a theorem to guarantee uniform global asymptotic stability (UGAS) and uniform local exponential stability (ULES) for a class of nonlinear non‐autonomous systems which includes passive systems. These properties (and a uniform local Lipschitz condition) guarantee robustness of stability while weaker properties, like uniform global stability plus global convergence, do not. Our main result is then used in the tracking control problem of mechanical systems and ships. We use an adaptive backstepping design and prove UGAS of the closed‐loop tracking error system, in particular, we obtain that both the tracking and parameter estimation errors converge uniformly globally to zero. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Exact distribution of edge-preserving MAP estimators for linear signal models with Gaussian measurement noise

We derive the exact statistical distribution of maximum a posteriori (MAP) estimators having edge-preserving nonGaussian priors. Such estimators have been widely advocated for image restoration and reconstruction problems. Previous investigations of these image recovery methods have been primarily empirical; the distribution we derive enables theoretical analysis. The signal model is linear with Gaussian measurement noise. We assume that the energy function of the prior distribution is chosen to ensure a unimodal posterior distribution (for which convexity of the energy function is sufficient), and that the energy function satisfies a uniform Lipschitz regularity condition. The regularity conditions are sufficiently general to encompass popular priors such as the generalized Gaussian Markov random field prior and the Huber prior, even though those priors are not everywhere twice continuously differentiable.