Smooth Lipschitz Research Articles

A note on 𝐶^{1,𝛼}-smooth approximation of Lipschitz functions

We show that on super-reflexive spaces a Moreau-Yosida type of regularisation by infimal convolution together with a known insertion-type theorem (a variant of Ilmanen’s lemma) easily give an approximation of a Lipschitz function by a C 1 , α C^{1,\alpha } -smooth Lipschitz function with the same Lipschitz constant. This is a generalisation of the well-known theorem of J.-M. Lasry and P.-L. Lions from Hilbert spaces. It also gives a new self-contained and probably simpler proof of the Lasry-Lions theorem.

Event‐triggered adaptive consensus control of nonlinear multi‐agent systems with unknown control directions and actuator saturation

AbstractThis article addresses the event‐triggered adaptive consensus control of nonlinear multi‐agent systems with unknown control direction and actuator saturation. A new robust adaptive control algorithm based on an event‐triggered mechanism is designed. The smooth Lipschitz function approximates the saturated nonlinear function, while the Nussbaum function handles unknown control directions and residual terms. The event‐triggered mechanism is designed to determine the time of communication, significantly reducing the communication burden. An additional estimator is utilized to deal with unknown parameters involved in neighbor dynamics and prevent information exchange to consistency errors between connected subsystems. The results show that all the signals of the closed‐loop system are uniformly bounded, and the consensus tracking error converges to a bounded set. Meanwhile, Zeno's behavior is eliminated. Simulation results confirm the superiority of the proposed method.

Distributed robust event-triggered adaptive control for a class of uncertain nonlinear multi-agent systems with actuator saturation

This paper studies the event-triggered consensus control problem for high-order uncertain nonlinear multi-agent systems with actuator saturation. By using a smooth Lipschitz function to approximate the saturation nonlinearity, an augment system and the Nussbaum function are adopted to deal with the residual terms of saturation nonlinearity based on adaptive backstepping method. Since excessive energy and communication resources will be consumed during the procedure to handle actuator saturation, two event-triggered mechanisms are proposed to save the communication resources and reduce the controllers’ update frequency. Whenever the triggered conditions are satisfied, the control signals transmitted to the actuators are updated and broadcasted to the neighboring area. A ’disturbance-like’ term is integrated so that the event-triggered control problem with actuator saturation can be transformed into a robust problem while the unknown disturbances are tackled by adaptive update laws. Moreover, the requirement for global communication topology known by all the agents is relaxed by introducing new estimators. All the signals in the closed-loop system are uniformly bounded and the consensus tracking errors are exponentially converged to a bounded set. Meanwhile, the Zeno behavior is excluded. Simulation results are employed to validate the advantages of our proposed methods.

Metric topology on the moduli space

<p>We define the smooth Lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology, which offers an answer to Tuschmann’s question.</p>

Learning Non-Parametric Models with Guarantees: A Smooth Lipschitz Regression Approach

We propose a non-parametric regression methodology that enforces the regressor to be fully consistent with the sample set and the ground-truth regularity assumptions. As opposed to the Nonlinear Set Membership technique, this constraint guarantees the attainment of everywhere differentiable surrogate models, which are more suitable to optimization-based controllers that heavily rely on gradient computations. The presented approach is named Smooth Lipschitz Regression (SLR) and provides error bounds on the prediction error at unseen points in the space. A numerical example is given to show the effectiveness of this method when compared to the other alternatives in a Model Predictive Control setting.

Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability

We consider concentrated vorticities for the Euler equation on a smooth domain Ω⊂R2 in the form ofω=∑j=1NωjχΩj,|Ωj|=πrj2,∫Ωjωjdμ=μj≠0, supported on well-separated vortical domains Ωj, j=1,…,N, of small diameters O(rj). A conformal mapping framework is set up to study this free boundary problem with Ωj being part of unknowns. For any given vorticities μ1,…,μN and small r1,…,rN∈R+, through a perturbation approach, we obtain such piecewise constant steady vortex patches as well as piecewise smooth Lipschitz steady vorticities, both concentrated near non-degenerate critical configurations of the Kirchhoff–Routh Hamiltonian function. When vortex patch evolution is considered as the boundary dynamics of ∂Ωj, through an invariant subspace decomposition, it is also proved that the spectral/linear stability of such steady vortex patches is largely determined by that of the 2N-dimensional linearized point vortex dynamics, while the motion is highly oscillatory in the 2N-codim directions corresponding to the vortical domain shapes.

One can hear the corners of a drum

We prove that the presence or absence of corners is spectrally determined in the following sense: any simply connected domain with piecewise smooth Lipschitz boundary cannot be isospectral to any connected domain, of any genus, which has smooth boundary. Moreover, we prove that amongst all domains with Lipschitz, piecewise smooth boundary and fixed genus, the presence or absence of corners is uniquely determined by the spectrum. This means that corners are an elementary geometric spectral invariant; one can hear corners.

Sliding mode control design via piecewise smooth Lipschitz surfaces based on contingent cone criteria

In order to improve flexibility of sliding mode control (SMC) for a class of nonlinear systems, a new control design method is proposed in this paper. The sliding surface is extended to be a generic Lipschitz continuous surface instead of a smooth one, with which different characteristics of sliding motion may be realized. Due to the nonsmoothness of the sliding surface, the control design problem is discussed in the framework of Filippov’s differential inclusion, by analyzing the geometric properties of the nonsmooth surface. Discontinuous control laws are presented based on new contingent cone criteria, which steer all of the trajectories of the closed-loop system to reach the piecewise smooth sliding surface in finite time.

Smooth approximations

We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f : X → Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C ( K ) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C 1 -smooth mappings together with their first derivatives. As a corollary we obtain new results on smooth approximation of C 1 -smooth mappings together with their first derivatives.

Formula omitted]-minimal [formula omitted]-regular stratification

Let M be an o -minimal structure expanding a real closed field R . We show that any definable set in R n can be stratified into cells, whose defining functions are C m smooth Lipschitz continuous functions with constant 2 n 3 / 2 , which have additional regularity conditions on the derivatives of higher order.

Smooth and Nonsmooth Lipschitz Controls for a Class of Vector Differential Equations

This paper is devoted to the study of a class of control problems associated to a nonlinear second-order vector differential equation with pointwise state constraints. The control is realized via a function of the state. We extend the results of Akkouchi, Bounabat, and Goebel to vector differential equations and furthermore consider the more general case. Under proper conditions, we prove the existence of optimal controls in the class of Lipschitz functions and obtain an optimality condition which looks somehow like the Pontryagin maximum principle for a smooth optimal control function. For a nonsmooth optimal control function, we derive a suboptimality condition by means of the Ekeland variational principle.

Smooth approximation of Lipschitz functions on Riemannian manifolds

We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous ε : M → ( 0 , + ∞ ) , and for every positive number r > 0 , there exists a C ∞ smooth Lipschitz function g : M → R such that | f ( p ) − g ( p ) | ⩽ ε ( p ) for every p ∈ M and Lip ( g ) ⩽ Lip ( f ) + r . Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville–Godefroy–Zizler's smooth variational principle.

C1-fine approximation of functions on Banach spaces with unconditional basis

We show that if X is a Banach space having an unconditional basis and a C p -smooth Lipschitz bump function, then for every C 1 -smooth function f from X into a Banach space Y , and for every continuous function e : X → (0, ∞), there exists a C p -smooth function g : X → Y such

Perturbed smooth Lipschitz extensions of uniformly continuous functions on Banach spaces

We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have C-p-smooth Lipschitz bump functions, and U is a bounded open subset of X, then, for every uniformly continuous function f : Y boolean AND U --> R and every epsilon > 0, there exists a C-p-smooth Lipschitz function F : X --> R such that |F(y)- f( y)| less than or equal to epsilon for every y is an element of Y boolean AND U.

Uniform and Lipschitz homotopy classes of maps

If X is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line R into X an element of H 1 (X, U/U 0 ), where U is the space of uniformly continuous functions from R to R and U 0 is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to H 1 (X, U/U 0 ) is surjective. If X is the n-dimensional torus, it is bijective, while if X is a compact orientable surface of genus > 1, it is not injective. In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds P to compact smooth manifolds X. With each such Lipschitz homotopy class we associate an element of H n (X, B + /B + 0 ), where n is the dimension of P, B is the space of bounded continuous functions from the positive real axis to R, and B + 0 is the set of all f ∈ B + such that lim t→ ∞ f(t) = 0.

On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space

We study the size of the sets of gradients of bump functions on the Hilbert space ℓ 2 , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in ℓ 2 can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space ℓ 2 can be uniformly approximated by C 1 smooth Lipschitz functions ψ so that the cones generated by the ranges of its derivatives ψ ' (ℓ 2 ) have empty interior. This implies that there are C 1 smooth Lipschitz bumps in ℓ 2 so that the cones generated by their sets of gradients have empty interior. On the other hand, we construct C 1 -smooth bounded starlike bodies A⊂ℓ 2 , which approximate the unit ball, so that the cones generated by the hyperplanes which are tangent to A have empty interior as well. We also explain why this is the best answer to the above questions that one can expect.

James' Theorem Fails for Starlike Bodies

Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump functions and to n-homogeneous polynomials on Banach spaces, and their geometrical properties are thus worth studying. In this paper we deal with the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C1 Lipschitz bump function if and only if there exists another C1 smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.

Null Sets and Essentially Smooth Lipschitz Functions

In this paper we extend the notion of a Lebesgue-null set to a notion which is valid in any completely metrizable Abelian topological group. We then use this definition to introduce and study the class of essentially smooth functions. These are, roughly speaking, those Lipschitz functions which are smooth (in each direction) almost everywhere.

A Chain Rule for Essentially Smooth Lipschitz Functions

In this paper we introduce a new class of real-valued locally Lipschitz functions (that are similar in nature and definition to Valadier's saine functions), which we call arcwise essentially smooth, and we show that if g : Rm \rightarrow R$ is arcwise essentially smooth on Rm and each function $f_j : R^n \rightarrow R, 1 \leq j \leq m$, is strictly differentiable almost everywhere in Rn, then $g \circ f$ is strictly differentiable almost everywhere in Rn, where $f \equiv (f_1,f_2,\ldots,f_m)$. We also show that all the semismooth and all the pseudoregular functions are arcwise essentially smooth. Thus, we provide a large and robust lattice algebra of Lipschitz functions whose generalized derivatives are well behaved.

Essentially Smooth Lipschitz Functions

In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability,D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipschitz functions that are integrable,D-representable and possess desirable differentiability properties. We also present some striking applications of our results to distance functions.