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Nonlocal boundary value problem of the first kind for an odinary linear second order differential equation with positive parameter at the highest derivative is considered. The existence and uniqueness, as well as, a uniformly stable estimate of classical solution is established under accurate condition on coefficients and location of nonlocal data carriers of multipoint boundary value condition. An essentiality of the revealed condition is confirmed by ill-posed problem examples.

In this note, we are concerned with uniform stability for a Leray–Hopf weak solution of the MHD equations and obtain that the Leray–Hopf weak solution is uniformly stable with respect to a small perturbation of initial velocity, magnetic and external forcing. This extends and improves the results given by Gallagher–Planchon and Dong–Jia for Navier–Stokes equations.

In this paper, we investigate the asymptotic stability of fractional resolvent families on Banach spaces and ordered Banach spaces. We show that an $$\alpha $$ -times resolvent family $$S_\alpha (t)$$ with generator A is uniformly Abel-stable if and only if $$0\in \rho (A)$$ , and if in addition $$S_\alpha (t)$$ is analytic and bounded, then $$S_\alpha (t)$$ is uniformly stable with $$\Vert S_\alpha (t)\Vert = O(t^{-\alpha })\, (t \rightarrow \infty )$$ . For a bounded positive $$\alpha $$ -times resolvent family on an ordered Banach space, we show that it cannot be uniformly stable if $$\alpha \in (1,2)$$ ; in the case of $$\alpha \in (0,1)$$ , $$0 \in \rho (A)$$ implies the same decay rate $$t^{-\alpha }$$ . Several results on strong stability are also given by using contour integrals, Tauberian theorems and subordination principles.

In this paper, a modified weak Galerkin (MWG) finite element method with weakly imposed boundary conditions is presented for solving convection-dominated diffusion equations. The method is shown uniformly stable for all diffusion parameters. The method converges at the optimal order for large diffusion problems in the energy norm, and at half a super-convergent order for small diffusion problems. Various numerical examples are presented, showing that the method is as effective as the weak Galerkin method.

We analyze Galerkin discretizations of a new well-posed mixed space–time variational formulation of parabolic partial differential equations. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space–time discretization methods introduced by Andreev (2013, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal., 33, 242–260) and by Steinbach (2015, Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math., 15, 551–566).

We study a symmetric three-level (in time) method with a weight and a symmetric vector two-level method for solving the initial-boundary value problem for a second-order hyperbolic equation with a small parameter $$\tau > 0$$ multiplying the highest time derivative, where the hyperbolic equation is a perturbation of the corresponding parabolic equation. It is proved that the solutions are uniformly stable in $$\tau $$ and time in two norms with respect to the initial data and the right-hand side of the equation. Additionally, the case where $$\tau $$ also multiplies the elliptic part of the equation is covered. The spacial discretization can be performed using the finite-difference or finite element method.

Benjamini et al. (Inst Hautes Etudes Sci Publ Math 90:5–43, 2001) showed that weighted majority functions of n independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability 𝜖, the probability p𝜖 that the weighted majority changes is at most C𝜖1∕4. They asked what is the best possible exponent that could replace 1∕4. We prove that the answer is 1∕2. The upper bound obtained for p𝜖 is within a factor of \(\sqrt {\pi /2}+o(1)\) from the known lower bound when 𝜖 → 0 and n𝜖 →∞.

Current studies demonstrate that uniformly stable indoor thermal environments do not completely ensure the thermal comfort of residents, and unreasonable temperature settings can cause discomfort and high energy consumption. Moreover, temperature ramps frequently occur in actual indoor environments. Thus, it is important to understand human thermal responses in moderate environments with temperature ramps to improve working conditions and reduce energy consumption. In this regard, an experiment using within−subjects design in a chamber was conducted to evaluate the effects of temperature ramps on human thermal responses. A total of 60 healthy participants (30 females and 30 males) were recruited to participate in the tests. Their subjective responses, thermal comfort and sensation, thermal acceptability and preference, humidity perception and facial thermal sensation were collected during the experiments. The results showed that temperature ramps significantly affected the thermal sensation, thermal comfort, and facial sensation in the ramp-down environments but not significantly in ramp-up environments. However, the effect of temperature ramp-down on the humidity sensation was only significant for the female participants. A 2 °C temperature ramp-up did not affect the thermal comfort of the participants and was acceptable for them. The male participants were more receptive to ambient temperatures than the female ones, whereas the female participants were more sensitive to cool environments than the male ones. Moreover, good fitting relationships between the thermal sensation and the thermal comfort, as well as the thermal sensation and the facial thermal sensation were observed. This study provides some references for the influence of temperature ramps on human thermal responses in actual indoor environments, and recommends temperature set points to reduce the use of cooling energy.

The problem of p-th moment stability for time-varying stochastic time-delay systems with Markovian switching is investigated in this paper. Some novel stability criteria are obtained by applying the generalized Razumikhin and Krasovskii stability theorems. Both p-th moment asymptotic stability and (integral) input-to-state stability are considered based on the notion and properties of uniformly stable functions and the improved comparison principles. The established results show that time-derivatives of the constructed Razumikhin functions and Krasovskii functionals are allowed to be indefinite, which improve the existing results on this topic. By applying the obtained results for stochastic systems, we also analyze briefly the stability of time-varying deterministic time-delay systems. Finally, examples are provided to illustrate the effectiveness of the proposed results.

In this well-researched book, Katherine Newman challenges the notion that the living environments of older people are uniformly stable and desirable. The premise of this book is that the life pathw...

In this paper we construct linear, uniformly stable, wavelet-like functions on arbitrary triangulations. As opposed to standard wavelets, only local orthogonality is required for the wavelet-like functions. Nested triangulations are obtained through refinement by two standard strategies, in which no regularity is required. One strategy inserts a new node at an arbitrary position inside a triangle and then splits the triangle into three smaller triangles. The other strategy splits two neighbouring triangles into four smaller triangles by inserting a new node somewhere on the edge between the triangles. In other words, non-uniform refinement is allowed in both strategies. The refinement results in nested spaces of piecewise linear functions. The detail-, or wavelet-spaces, are made to satisfy certain orthogonality conditions which locally correspond to vanishing linear moments. It turns out that this construction is uniformly stable in the L norm, independently of the geometry of the original triangulation and the refinements.

A nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

This paper investigates the problem of model reduction for a class of switched linear parameter varying (LPV) systems with persistent dwell-time (PDT) switching property. The PDT switching generalizes the dwell-time and average dwell-time switching that are commonly concerned in the existing literature. To reduce the original high-order model with less conservatism, the constructed reduced-order model is both parameterized and semi-time-varying such that the model error system is uniformly stable over the PDT switching and guarantees an error performance in H ∞ sense. A numerical example is used to verify the validity and advantage of the theoretical results.

Using the asymmetric discretization technique, an explicit finite difference scheme is constructed for one-dimensional spatial fractional diffusion equations (FDEs). The spatial fractional derivative is approximated by the weighted and shifted Grünwald difference operator. The scheme can be solved explicitly by calculating unknowns in the different nodal-point sequences at the odd time-step and the even time-step. The uniform stability is proven and the error between the discrete solution and analytical solution is theoretically estimated. Numerical examples are given to verify theoretical analysis.

The main results of the paper are generalizations of the Razumikhin and of the Krasovskii classical stability theorems for stability analysis of time-varying time-delay systems. The condition of negativity of the time-derivative of Razumikhin functions and Krasovskii functionals is weakened. This is achieved by using the notion and properties of uniformly stable functions. We also show how to apply the results to the stability analysis of linear time-varying time-delay systems of retarded type. Both the system matrices and time-delays are allowed to be time-varying. Some constructive sufficient stability conditions are obtained and their effectiveness is demonstrated by some examples.

It is shown that there exist linear nonautonomous systems of any dimension that have unbounded coefficients and possess the properties of residual uniform stability and asymptotic stability, but are not uniformly stable.

A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discretel2norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.

International audience; In this paper, we consider triangular nonconforming finite element approximations of an interface elliptic problem. We propose two extensions of the conforming Nitsche's extended finite element method to the nonconforming case. The first one is obtained by adding stabilisation terms on the cut edges, and the second one by modifying the Crouzeix-Raviart basis functions on the cut cells. Both discrete problems are uniformly stable and yield optimal a priori error estimates, uniformly with respect to the diffusion parameters. Moreover, we show that they exhibit the same robustness with respect to the position of the interface as the classical conforming method. We then validate these results numerically. Finally, we propose a nonconforming approximation of the interface Stokes problem based on the modified Crouzeix-Raviart elements and we illustrate it numerically. c; Dans cet article, nous considérons les approximations par éléments finis triangulaires non conformes d'un problème elliptique d'interface. Nous proposons deux extensions de la méthode d'éléments finis étendue de Nitsche conforme au cas non conforme. La première est obtenue en ajoutant des termes de stabilisation sur les bords coupés, et la seconde en modifiant les fonctions de base de Crouzeix-Raviart sur les cellules coupées. Les deux problèmes discrets sont uniformément stables et donnent des estimations d'erreur a priori optimales, uniformément par rapport aux paramètres de diffusion. De plus, nous montrons qu'ils présentent la même robustesse vis-à-vis de la position de l'interface que la méthode classique de conformation. Nous validons ensuite ces résultats numériquement. Enfin, nous proposons une approximation non-conforme du problème de Stokes de l'interface basée sur les éléments de Crouzeix-Raviart modifiés et nous l'illustrons numériquement.

We prove a necessary and sufficient condition for the existence of Lyapunov density for a system of coupled autonomous ordinary differential equations. In particular, we characterize the kinds of couplings that preserve almost everywhere uniform stability of the origin provided the isolated systems have an almost everywhere uniformly stable equilibrium point at the origin.

We consider a one-parameter family of difference schemes approximating a nonlocal heat problem with variable coefficient. We study the spectral properties of the main difference operator of the scheme. An energy norm in which the schemes are uniformly stable is defined on the space of grid functions. The corresponding stability condition is derived.

In this paper, a generalized Volterra–Levin equation is considered. 3/2-stability conditions are established for ensuring that the zero solution of the equation is uniformly stable, asymptotically stable and globally asymptotically stable. Further results are obtained under nonnegativity assumption. The main results are new and different from the known results.

The sensorless high speed-tracking control problem for a surface-mount permanent magnet synchronous motor via stator currents and voltages measurements is tackled, by interconnecting an H ∞ -controller and a reduced order sliding-mode observer. The observer-control scheme is robust against external load disturbances. The rotor position and speed variables are estimated in finite time by means of a reduced order observer. Thus, an output-feedback H ∞ -controller is designed such that the undisturbed system is uniformly stable around the desired speed reference, whereas the effects of the disturbances are attenuated. The feasibility of the proposed robust sensorless controller is supported by numerical simulations.

sic) Abstract. In this paper we are concerned with a nonlocal problem of a coupled system of dif- ferential equations. We study the local existence of the solution and its continuous dependence. The global existence and its uniform stability is being studied.

The main result of this paper states that phase retrieval in infinite-dimensional Hilbert spaces is never uniformly stable, in sharp contrast to the finite-dimensional setting in which phase retrieval is always stable. This leads us to derive stability results for signals depending on how well they are approximated by finite expansions.

A stabilization/observability estimate and asymptotic energy decay rates are derived for a wave equation with nonlinear damping in a portion of the interior and Wentzell condition on the boundary: ∂ ν u + u = Δ T u \partial _{\nu } u + u = \Delta _{T}u . The dissipation does not affect a full collar of the boundary, thus leaving out a portion subjected to the high-order Wentzell condition. Observability of wave equations with damping supported away from the Neumann boundary is known to be intrinsically more difficult than the corresponding Dirichlet problem because the uniform Lopatinskii condition is not satisfied by such a system. In the case of a Wentzell boundary, the situation is more difficult since the “natural” energy now includes the H 1 H^{1} Sobolev norm of the solution on the boundary. To establish uniform stability it is necessary not only to overcome the presence of the Neumann boundary operator, but also to establish an inverse-type coercivity estimate on the H 1 H^{1} trace norm of the solution. This goal is attained by constructing multipliers based on a refinement of nonradial vector fields employed for “unobserved” Neumann conditions. These multipliers, along with a suitable geometry (local convexity), allow reconstruction of the high-order part of the potential energy from the damping that is supported only in a far-off region of the domain.

In this article, we consider rectangular finite element methods for fourth order elliptic singular perturbation problems. We show that the non-C 0 rectangular Morley element is uniformly convergent in the energy norm with respect to the perturbation parameter. We also propose a C 0 extended high order rectangular Morley element and prove the uniform convergence. Finally, we do some numerical experiments to confirm the theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 00: 000–000, 2012

The main aim of this note is to construct topological conjugacies between linear nonautonomous contractions with discrete time. As a consequence, we obtain topological conjugacies between linear and nonlinear nonautonomous contractions. We consider the general case of linear contractions with respect to arbitrary growth rates e - c ρ ( m ) . This includes the usual exponential contractions with ρ ( m ) = m as a very special case. We also consider the case of contractions that are not uniformly stable. In addition, we show that all conjugacies are locally Holder outside the origin, and locally Holder everywhere for a large class of growth rates ρ ( m ) .

In this paper, we study the existence of a uniformly stable positive monotonic solution for the nonlocal Cauchy problem with the nonlocal condition where

In this paper, a theoretical treatment of the stability behavior of the zero solution of nonlinear damped oscillator in the vectorial case is investigated. We study the sufficient conditions for the boundedness of solution of the nonlinear damped vectorial oscillator and the conditions for the stability of the zero solution to be uniformly stable as well as asymptotically stable.

This paper deals with the stability problems of nonautonomous impulsive differential systems with time-delay. By utilizing the method of mathematic induction, proofs by contradiction and Lyapunov-Krasovskii functional, we obtain several uniformly stable criteria about the linear and nonlinear nonautonomous impulsive systems with time delay. One numerical example and its simulations are given to illustrate the effectiveness of the theoretical result.

In this work we deal with the problem of the stability and uniform stability of the perturbed n -dimensional Volterra integral and differential equation x ′ ( t ) = A ( t ) x ( t ) + ∫ − ∞ t C ( t , s ) x ( s ) d s + ∫ − ∞ t D ( t , s ) x ′ ( s ) d s + b ( t ) . Some sufficient conditions for the zero solution of this equation to be stable as well as uniformly stable have been obtained.

In this paper, we consider the Cauchy problem of the Boltzmann equation with potential force in the whole space. When some more natural assumptions compared with those of the previous works are made on the potential force, we can still obtain a unique global solution to the Boltzmann equation even for the hard potential cases by energy method, if the initial data are sufficiently close to the steady state. Moreover, the solution is uniformly stable. Copyright © 2010 John Wiley & Sons, Ltd.

We consider a family of quadrilateral or hexahedral mixed hp -finite elements for an incompressible flow problem with Q r -elements for the velocity and discontinuous -elements for the pressure where the order r can vary from element to element between 2 and an arbitrary bound. For multilevel adaptive grids with hanging nodes and a sufficiently small mesh size, we prove the inf-sup condition uniformly with respect to the mesh size and the polynomial degree.

In this paper, we give some sufficient conditions for the zero solution of an m-dimensional delay difference equation of the form to be uniformly stable and asymptotically stable.

A class of new mixed formulations for elasticity is developed and analysed. The formulations are based on the discrete evss method, introduced in the context of incompressible viscoelastic flows by A. Fortin, M. Fortin and co-workers. A key feature is a stabilization term that renders coercive a problem that might not otherwise be so. The focus in this work is on behaviour in the incompressible limit and the goal is that of obtaining formulations that are uniformly stable and convergent. Concrete examples are presented of element choices that lead to unstable formulations in the classical formulation, and which are stable for the formulations introduced here. A selection of numerical results illustrates in a comparative way the behaviour of the elements introduced.

We prove here some mapping theorems for the operators of class Cp (p > 0) on Hilbert spaces defined by B. Sz-Nagy and C. Foias in [12]. More precisely, we answer the following question: What can be said about membership of f(T) in the classes Cp (H) when T belongs to a given one of them and f is in the disc algebra? As a corollary, we recover in this way the famous power inequality. We also introduce a Harnack ordering relation between such operators, we define the corresponding Harnack parts in the class Cp, and we give some results related to uniformly stable operators. These parts involve some operatorial Harnack inequalities as well as von Neumann inequalities, which generalize the Harnack inequalities for contractions given by C. Foias [8], K. Fan [5] and I. Suciu [17]. Finally, we define the Harnack and hyperbolic metrics, and we prove that these are complete on the set of all uniformly stable operators, respectively on the Harnack parts of such operators.

A conformal finite-difference time-domain algorithm for the solution of electrodynamic problems in general perfectly conducting 3D geometries is presented. Unlike previous conformal approaches it has the second-order convergence without the need to reduce the maximal stable time step of conventional staircase approach. A novel proof for the local error rate for general geometries is given, and the method is verified and compared to other approaches by means of several numerical 2D examples. Copyright © 2003 John Wiley & Sons, Ltd.