Lipschitz Perturbations Research Articles

О МНОЖЕСТВАХ МЕТРИЧЕСКОЙ РЕГУЛЯРНОСТИ ОТОБРАЖЕНИЙ В ПРОСТРАНСТВАХ С ВЕКТОРНОЗНАЧНОЙ МЕТРИКОЙ

Spaces with vector-valued metric are considered. The values of a vectorvalued metric are elements of a cone in some linear normed space. The concept of the set of metric regularity for mapping in spaces with vector-valued metric is formulated. A statement on the stability of the set of metric regularity of a given mapping for its Lipschitz perturbations in spaces with vector-valued metric is obtained.

Fixed-Time Homogeneous Integral Controller

Abstract We propose a homogeneous integral controller with positive homogeneity degree, for systems with relative degree two and three. Its design and the convergence proof is achieved by the construction of an explicit homogenous Lyapunov function. A combination of this controller, with a discontinuous one developed by the authors in a previous work, allows to achieve: (i) fixed time convergence, that is that the maximal convergence time is a constant independent of the size of the initial conditions; (ii) insensitivity to Lipschitz (matched) perturbations, i.e. perturbations with bounded derivative; and (iii) providing a continuous control signal, and thus reducing the chattering effect of a discontinuous controller.

Approximation and convergence rate of nonlinear eigenvalues: Lipschitz perturbations of a bounded self-adjoint operator

We consider nonlinear eigenvalue problems of the form (⁎)Tx+ϵB(x)=λx, where T is a self-adjoint bounded linear operator acting in a real Hilbert space H, and B:H→H is a (possibly) nonlinear continuous perturbation term. Assuming that λ0 is an isolated eigenvalue of finite multiplicity of T, we ask if for ϵ≠0 and small there are “eigenvalues” of (⁎) near λ0, that is, numbers λϵ for which (⁎) is satisfied by some normalized “eigenvector” xϵ of T+ϵB. In this paper we recall some recent results giving an affirmative answer to this question, and for these cases we prove – assuming in addition Lipschitz continuity on B – upper and lower bounds for the perturbed eigenvalues λϵ which are determined by those for the nonlinear Rayleigh quotient 〈B(v),v〉/〈v,v〉 with v in the eigenspace Ker(T−λ0I). This yields in particular information on the rate of convergence of λϵ to λ0 as ϵ→0. Applications are given in the sequence space l2, and in the Sobolev space H01 to deal with some nonlinearly perturbed ordinary or partial differential equations.

Dynamical poroplasticity model – Existence theory for gradient type nonlinearities with Lipschitz perturbations

In this article we study existence theory to a non-coercive fully dynamic model of poroplasticity with the non-homogeneous boundary conditions where the constitutive function is a continuous element of class LM (it is a sum of a maximal monotone map G and a globally Lipschitz map l). Without any additional growth conditions we are able to prove the existence of a solution such that the inelastic constitutive equation is satisfied in the measure-valued sense. Moreover, if G is a gradient of a differentiable convex function, then there exists a solution such that the constitutive equation is satisfied almost everywhere.

Lipschitz perturbations of a class of approximately controllable linear systems

This paper is devoted to the analysis of an approximately controllable system perturbed by a certain Lipschitz-continuous nonlinearity. By assuming that the intensity of the nonlinear influence is ‘weak’, we prove that the perturbed system is still approximately controllable. Using this perturbation theory, we prove that a third-order semilinear dispersion equation on a finite interval is approximately controllable whenever the nonlinearity effect is ‘weak’. The result in the paper is a complement of the results obtained in (Zhou in SIAM J. Control Optim. 21:551-565, 1983), in which the control operator (resp. infinitesimal generator of the dynamics) of the unperturbed linear system is assumed to be bounded (resp. generated a differentiable semigroup on the state space).

On the Continuity of Inverse Mappings for Lipschitz Perturbations of Covering Mappings

In this paper, we study the question of the existence of a continuous right inverse mapping for a covering mapping. To describe covering mappings that have a continuous right inverse, a concept of strong covering is introduced. It is shown that the property of strong covering is stable under small Lipschitz perturbations.

Topological conjugacy for Lipschitz perturbations of non-autonomous systems

In this paper, topological conjugacy for two-sided non-hyperbolic and non-autonomous discrete dynamical systems is studied. It is shown that if the system has covering relations with weak Lyapunov condition determined by a transition matrix, there exists a sequence of compact invariant sets restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by the transition matrix. Moreover, if the systems have covering relations with exponential dichotomy and small Lipschitz perturbations, then there is a constructive verification proof of the weak Lyapunov condition, and so topological dynamics of these systems are fully understood by symbolic representations. In addition, the tolerance of Lipschitz perturbation can be characterised by the dichotomy tuple . Here, the weak Lyapunov condition is adapted from [12,24,15] and the exponential dichotomy is from [2].

Perturbations of vectorial coverings and systems of equations in metric spaces

E. R. Avakov, A. V. Arutyunov, S. E. Zhukovskiĭ, and E. S. Zhukovskiĭ studied the problem of Lipschitz perturbations of conditional coverings of metric spaces. Here we propose some extension of the concept of conditional covering to vector-valued mappings; i.e., the mappings acting in products of metric spaces. The idea is that, to describe a mapping, we replace the covering constant by the matrix of covering coefficients of the components of the vector-valued mapping with respect to the corresponding arguments. We obtain a statement on the preservation of the property of conditional and unconditional vectorial coverings under Lipschitz perturbations; the main assumption is that the spectral radius of the product of the covering matrix and the Lipschitz matrix is less than one. In the scalar case this assumption is equivalent to the traditional requirement that the covering constant be greater than the Lipschitz constant. The statement can be used to study various simultaneous equations. As applications we consider: some statements on the solvability of simultaneous operator equations of a particular form arising in the problems on n-fold coincidence points and n-fold fixed points; as well as some conditions for the existence of periodic solutions to a concrete implicit difference equation.

On uniqueness of weak solutions for the thin-film equation

In any number of space variables, we study the Cauchy problem related to the fourth-order degenerate diffusion equation ∂sh+∇⋅(h∇Δh)=0. This equation, derived from a lubrication approximation, also models the surface tension dominated flow of a thin viscous film in the Hele-Shaw cell. Our focus is on uniqueness of weak solutions in the “complete wetting” regime, when a zero contact angle between liquid and solid is imposed. In this case, we transform the problem by zooming into the free boundary and look at small Lipschitz perturbations of a quadratic stationary solution. In the perturbational setting, the main difficulty is to construct scale invariant function spaces in such a way that they are compatible with the structure of the nonlinear equation. Here we rely on the theory of singular integrals in spaces of homogeneous type to obtain linear estimates in these functions spaces which provide “optimal” conditions on the initial data under which a unique solution exists. In fact, this solution can be used to define a class of functions in which the original initial value problem has a unique (weak) solution. Moreover, we show that the interface between empty and wetted regions is an analytic hypersurface in time and space.

Lipschitz perturbations of expansive systems

We extend some known results from smooth dynamical systems to the category of Lipschitz homeomorphisms of compact metric spaces. We consider dynamical properties as robust expansiveness and structural stability allowing Lipschitz perturbations with respect to a hyperbolic metric. We also study the relationship between Lipschitz topologies and the $C^1$ topology on smooth manifolds.

Local stability and parameter dependence of mild solutions for stochastic differential equations

For nonlinear stochastic equations dx(t )=( Ax(t )+ f (t, x(t), λ)) dt + g(t, x(t), λ) dω(t )w ith parameter λ in a Hilbert space, we show the existence and uniqueness of mild solutions. Provided that f satisfies a locally Lipschitz condition and g is a uniformly Lipschitz function, some sufficient conditions for p (p ≥ 2) moment locally exponential stability as well as almost surely exponential stability of mild solutions are obtained under a sufficiently small initial value ξ . Meanwhile, we also consider parameter dependence of stable mild solutions for the stochastic system if f , g are sufficiently small Lipschitz perturbations in the parameter λ.

Quasi-linear Compressed Sensing

Inspired by significant real-life applications, in particular, sparse phase retrieval and sparse pulsation frequency detection in Asteroseismology, we investigate a general framework for compressed sensing, where the measurements are quasi-linear. We formulate natural generalizations of the well-known Restricted Isometry Property (RIP) towards nonlinear measurements, which allow us to prove both unique identifiability of sparse signals as well as the convergence of recovery algorithms to compute them efficiently. We show that for certain randomized quasi-linear measurements, including Lipschitz perturbations of classical RIP matrices and phase retrieval from random projections, the proposed restricted isometry properties hold with high probability. We analyze a generalized Orthogonal Least Squares (OLS) under the assumption that magnitudes of signal entries to be recovered decay fast. Greed is good again, as we show that this algorithm performs efficiently in phase retrieval and asteroseismology. For situations where the decay assumption on the signal does not necessarily hold, we propose two alternative algorithms, which are natural generalizations of the well-known iterative hard and soft-thresholding. While these algorithms are rarely successful for the mentioned applications, we show their strong recovery guarantees for quasi-linear measurements which are Lipschitz perturbations of RIP matrices.

Semilinear observation systems

In this paper, we introduce locally Lipschitz observation systems for nonlinear semigroups and show that they can be represented by an ‘admissible’ nonlinear output operator defined on a suitable subspace. In the semilinear case, this concept fits well to the Lebesgue extension known from linear system theory. For semilinear systems, we show robustness of exact observability near equilibria under locally small Lipschitz perturbations. Finally, we apply our results to a damped nonlinear plate equation and a semilinear thermo-elastic system.

Parameter dependence of stable manifolds for nonuniform (µ, ν)-dichotomies

We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x′ = A(t)x + f(t, x, λ), t ≠ τi and x(τi+) = Bix(τi) + gi(x(τi), λ), i ∈ ℕ in Banach spaces, assuming that the linear impulsive differential equation x′ = A(t)x, t ≠ τi and x(τi+) = Bix(τi), i ∈ ℕ admits a nonuniform (µ, ν)-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter λ and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.

On Lipschitz Perturbations of a Self-Adjoint Strongly Positive Operator

In this paper we study semilinear equations of the formAu+λF(u)=f, whereAis a linear self-adjoint operator, satisfying a strong positivity condition, andFis a nonlinear Lipschitz operator. As applications we develop Krasnoselskii and Ky Fan type approximation results for certain pair of maps and to illustrate the usability of the obtained results, the existence of solution of an integral equation is provided.

Bifurcations from nondegenerate families of periodic solutions in Lipschitz systems

The paper addresses the problem of bifurcation of periodic solutions from a normally nondegenerate family of periodic solutions of ordinary differential equations under perturbations. The approach to solve this problem can be described as transforming (by a Lyapunov–Schmidt reduction) the initial system into one which is in the standard form of averaging, and subsequently applying the averaging principle. This approach encounters a fundamental problem when the perturbation is only Lipschitz (nonsmooth) as we do not longer have smooth Lyapunov–Schmidt projectors. The situation of Lipschitz perturbations has been addressed in the literature lately and the results obtained conclude the existence of the bifurcated branch of periodic solutions. Motivated by recent challenges in control theory, we are interested in the uniqueness problem. We achieve this in the case when the Lipschitz constant of the perturbation obeys a suitable estimate.

Robust nonuniform dichotomies and parameter dependence

We establish the robustness of linear cocycles with an exponential dichotomy, under sufficiently small Lipschitz perturbations, in the sense that the existence of an exponential dichotomy for a given cocycle persists under these perturbations. We consider cocycles in Banach spaces, as well as the general case of nonuniform exponential dichotomies, and also the general case of an exponential behavior e c ρ ( n ) , given by an arbitrary sequence ρ ( n ) including the usual exponential behavior ρ ( n ) = n as a very special case. Moreover, we show that the projections of the exponential dichotomies obtained from the perturbation vary continuously with the parameter, and in fact that they are locally Lipschitz on finite-dimensional parameters.

Locally Lipschitz perturbations of bisemigroups

In this work we study the ill posed semilinear system $\dot{x}= Lx + f(\xi,t), \dot{y}= Ry + g(\xi,t)$, $\xi=(x,y)$, in Banach spaces where $L$ and $R$ are the infinitesimalgenerators of two $C_o$ semigroups $\{\mathcal{L}(t), t\geq 0\}$ and $\{\mathcal{R}(-t), t\geq 0\}$ respectively.The nonlinearity $h=(f,g)$ is continuous in $t$ and locally Lipschitz continuous in $\xi$ locally uniformly in $t$. $\cdot $ We show the existence and uniqueness of what we call local dichotomous mild solutions (DMS) that take the form$x(t) = e^{(t-t_1)L}x_1 + \int_{t_1}^{t} e^{(t-s)L} f(\xi(s), s)ds$$ y(t)= e^{-(t_2-t)R} y_2 - \int_{t}^{t_2} e^{-(s-t)R} g(\xi(s), s) ds$$ t_1 for any sufficiently small time interval $[t_1, t_2]$ and any given $\xi :=(x_1, y_2)$ in a sufficiently small neighbourhood. $\cdot $ We show that in the uniform $C^0$-norm DMSs vary continuously with $[t_1, t_2]$ and Lipschitz-continuously with $\xi $. $\cdot $ We study the regularity of DMSs under various hypotheses. $\cdot $ A simple example that leads to a bisemigroup is a semilinearelliptic system that arises whenconsidering solitary waves in an infinite cylinder:$u_{x x}+\Delta u = f(u), \quad u|_{\Gamma} = 0, \quad\Gamma= \mathbb{R}\times \partial\Omega, \quad (x, y, u)\in \mathbb{R}\times \Omega\times\mathbb{R}^mwhere $\Omega$ is a bounded region in $ \mathbb{R}^n$ with $C^2$ boundary and $\Delta$ is the Laplacian in the variable $y\in \Omega$.

A Grobman–Hartman theorem for general nonuniform exponential dichotomies

For a nonautonomous dynamics with discrete time given by a sequence of linear operators A m , we establish a version of the Grobman–Hartman theorem in Banach spaces for a very general nonuniformly hyperbolic dynamics. More precisely, we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates e c ρ ( n ) , determined by a sequence ρ ( n ) . For all sufficiently small Lipschitz perturbations A m + f m we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators A m . We also show that all conjugacies are Hölder continuous. We note that the usual exponential behavior is included as a very special case when ρ ( n ) = n , but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when ρ ( n ) = log n .

Covering mappings and their applications to differential equations unsolved for the derivative

We continue to study the properties of covering mappings of metric spaces and present their applications to differential equations. To extend the applications of covering mappings, we introduce the notion of conditionally covering mapping. We prove that the solvability and the estimates for solutions of equations with conditionally covering mappings are preserved under small Lipschitz perturbations. These assertions are used in the solvability analysis of differential equations unsolved for the derivative.