Semigroups Of Operators Research Articles

Heat Kernels Estimates for Hermitian Line Bundles on Manifolds of Bounded Geometry

We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian upper bound for the associated heat kernel. The proof is based on some tools from the theory of operator semigroups in a Hilbert space, results on Sobolev spaces adapted to the current setting, and weighted estimates with appropriate exponential weights.

Trajectory Controllability of Dynamical Systems with Non-instantaneous Impulses

This manuscript considered the system governed by the non-instantaneous impulsive evolution control system and discusses trajectory controllability of the governed system with classical and nonlocal initial conditions over the general Banach space. The results of the trajectory controllability for governed systems are obtained through the concept of operator semigroup and Gronwall’s inequality. This manuscript is also equipped with examples to illustrate the applications of derived results.

A Factorization of a Quadratic Pencils of Accretive Operators and Applications

A canonical factorization is given for a quadratic pencil of accretive operators in a Hilbert space. We establish a criterion in order that the linear factors, into which the pencil splits, generates a holomorphic semi-group of contraction operators. As an application, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second-order differential equation in the non-homogeneous case. An illustrative example is also given.

One-dimensional Wiener process with the properties of partial reflection and delay

In this paper, we construct the two-parameter semigroup of operators associated with a certain one-dimensional inhomogeneous diffusion process and study its properties. We are interested in the process on the real line which can be described as follows. At the interior points of the half-lines separated by a point, the position of which depends on the time variable, this process coincides with the Wiener process given there and its behavior on the common boundary of these half-lines is determined by a kind of the conjugation condition of Feller-Wentzell's type. The conjugation condition we consider is local and contains only the first-order derivatives of the unknown function with respect to each of its variables.
 The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding conjugation problem for a second order linear parabolic equation to which the above problem is reduced. Its classical solvability is obtained by the boundary integral equations method under the assumption that the initial function is bounded and continuous on the whole real line, the parameters characterizing the Feller-Wentzell conjugation condition are continuous functions of the time variable, and the curve defining the common boundary of the domains is determined by the function which is continuously differentiable and its derivative satisfies the Hölder condition with exponent less than $1/2$.

A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusions of order [formula omitted] with delay

In this paper, we formulate a new set of sufficient conditions for the approximate controllability of fractional evolution stochastic integrodifferential delay inclusions of order r∈(1,2) with nonlocal conditions in Hilbert space. Martelli’s fixed point theorem, multivalued functions, cosine and sine families, fractional calculus and operator semigroups are used to establish the results under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the applicability of the obtained theoretical results.

Trajectory Controllability of the Systems Governed by Hilfer Fractional Systems

In this manuscript, we consider a nonlinear system governed by Hilfer fractional integro-differential equations in a Banach space. Using the concept of operator semigroup and Gronwall’s inequality, we have established the trajectory controllability of the integro-differential equation with local and non-local conditions. Finally, we have given an example to illustrate the application of the derived results

AN EXPLICIT CHARACTERIZATION OF THE DOMAIN OF THE INFINITESIMAL GENERATOR OF A SYMMETRIC DIFFUSION SEMIGROUP ON LP OF A COMPLETE POSITIVE SIGMA-FINITE MEASURE SPACE

Let X be a complete positive σ–finite measure space and {At}t≥0 be a symmetric diffusion semigroup of contraction operators on Lp(X). We prove that for 1<p<∞, the domain of the infinitesimal generator of the semigroup is precisely the space ∫01Ash ds:h∈Lp(X). We also establish that for 1<p<∞, the function spaces 2n-1∫02-(n-1)Ash ds|h∈Lp(X) are equal for every n∈ℤ.

Approximate controllability of a parabolic integro-differential equation with nonconstant memory kernel under a weak constraint condition

The approximate controllability of a parabolic integro-differential equation system has already been shown for the case where the memory kernel is a constant. However, when this condition on the memory kernel and considering instead only the weak constraint of boundedness, the problem becomes more complex and it is less easy to prove the approximate controllability of the system. In this study, based on the operator semigroup method, a detailed proof of the approximate controllability of the system is given for the case where the memory kernel is nonconstant and bounded. In addition, three numerical simulation studies are performed to verify the theoretical proof.

Weighted Composition Semigroups on Some Banach Spaces

We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis (Linear Algeb Appl 84:359–371, 1986) and König (Michigan Math J 37:469–476, 1990) and answers a question of Siskakis, A. G. proposed in Siskakis (Contemp Math 213:229–252, 1998). We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication operator \(M_z\).

Uniform convergence of operator semigroups without time regularity

When we are interested in the long-term behaviour of solutions to linear evolution equations, a large variety of techniques from the theory of C_0-semigroups is at our disposal. However, if we consider for instance parabolic equations with unbounded coefficients on {mathbb {R}}^d, the solution semigroup will not be strongly continuous, in general. For such semigroups many tools that can be used to investigate the asymptotic behaviour of C_0-semigroups are not available anymore and, hence, much less is known about their long-time behaviour. Motivated by this observation, we prove new characterisations of the operator norm convergence of general semigroup representations—without any time regularity assumptions—by adapting the concept of the “semigroup at infinity”, recently introduced by M. Haase and the second named author. Besides its independence of time regularity, our approach also allows us to treat the discrete-time case (i.e. powers of a single operator) and even more abstract semigroup representations within the same unified setting. As an application of our results, we prove a convergence theorem for solutions to systems of parabolic equations with the aforementioned properties.

Semigroups of Composition Operators and Integral Operators in BMOA-Type Spaces

The aim of this article is to study semigroups of composition operators \(T_t=f\circ \phi _t\) on the BMOA-type spaces \(\textit{BMOA}_p\), and on their “little oh” analogues \(\textit{VMOA}_p\). The spaces \(\textit{BMOA}_p\) were introduced by R. Zhao as part of the large family of F(p, q, s) spaces, and are the Möbius invariant subspaces of the Dirichlet spaces \(D^p_{p-1}\). We study the maximal subspace \([\phi _t, \textit{BMOA}_p]\) of strong continuity, providing a sufficient condition on the infinitesimal generator of \(\{\phi _t\}\), under which \([\phi _t, \textit{BMOA}_p]=\textit{VMOA}_p\), and a related necessary condition in the case where the Denjoy–Wolff point of the semigroup is in \({{\mathbb {D}}}\). Further, we characterize those semigroups, for which \([\phi _t, \textit{BMOA}_p]=\textit{VMOA}_p\), in terms of the resolvent operator of the infinitesimal generator of \((T_t|_{\textit{VMOA}_p})\). In addition we provide a connection between the maximal subspace of strong continuity and the Volterra-type operators \(T_g\). We characterize the symbols g for which \(T_g:\,\textit{BMOA}\rightarrow \textit{BMOA}_1\) is bounded or compact, thus extending a related result to the case \(p=1\). We also prove that for \(1<p<2\) compactness of \(T_g\) on \(\textit{BMOA}_p\) is equivalent to weak compactness.

Idempotent uninorms and nullnorms on bounded posets

The paper deals with uninorms and nullnorms as basic semi-group operations which are commutative and monotone (increasing). These operations were first introduced on the unit interval and later generalized to bounded lattices. In [Kalina 2019], they were introduced on bounded posets. This contribution is a generalization and extension of the results in [Kalina 2019]. Some necessary and some sufficient conditions for the existence of idempotent uninorms and idempotent nullnorms on bounded posets are studied. Finally, some application examples are provided.

Stabilization of time delay systems with saturations via PDE predictor boundary control design

This paper addresses the stabilization issue of linear time delay system with input saturation and distinct input delays via predictor feedback boundary control algorithm by employing transport partial differential equations (PDEs). First, the addressed ordinary differential equation (ODE) system with input delay is equivalently represented as a cascade of an ODE and transport PDEs. Second, by employing the backstepping Volterra integral transformation technique, the equivalent cascade system is transformed into a stable target system, whose kernels are solved by the constraints satisfying transport PDEs. Third, based on the boundary conditions of the obtained invertible transformation, the proposed feedback control law can be formulated. Fourth, by applying semigroup operator theory, the well-posedness of the resulting system is proved and consequently, novel exponential stability conditions of the addressed system are established. Then, the domain of attraction region under the given actuator saturation constraints is estimated with the help of the solution of obtained stability conditions. Finally, a demonstrative simulation example is offered to verify the feasibility and usefulness of the results.

The Cauchy–Dirichlet problem for the Moore–Gibson–Thompson equation

The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation is analyzed. With the focus on non-homogeneous boundary data, two approaches are offered: one is based on the theory of hyperbolic systems, while the other one uses the theory of operator semigroups. Both methods have in common that known results for the Dirichlet problem for the second-order wave equation are instrumental. The obtained interior and boundary regularity estimates are optimal.

New observer-based boundary control approaches to vibration reduction and disturbance attenuation of a flexible arm

This paper addresses the vibration reduction and disturbance attenuation problems of a flexible arm, which is modelled as a fourth-order partial differential equation. The main control objectives lie in reducing the influence of the disturbance, steering the flexible arm to the desired angular position and suppressing the vibration, simultaneously. To achieve the objectives, firstly, two different disturbance observers are designed. Secondly, we construct boundary controllers based on the two designed disturbance observers and boundary measurable signals. Then, with the two disturbance observers, the boundary disturbance compensation is achieved, and with the designed boundary controllers, the uniform boundedness of the flexible arm is guaranteed. Moreover, the well-posedness of the closed-loop system under the designed boundary controllers is discussed by means of the operator semigroup theory. Finally, the effectiveness and advantages of the proposed controllers are demonstrated through simulations and physical experiments.

Controllability of neutral impulsive fractional differential equations with Atangana-Baleanu-Caputo derivatives

In this article, we aim to establish the controllability results for fractional differential equations of neutral type with Atangana-Baleanu-Caputo derivatives. We develop these results with the help of the theory of semigroup operators and fixed point theorem coupled with a measure of noncompactness. An example is also presented to verify the applicability of the obtained results.

Dual-phase-lag one-dimensional thermo-porous-elasticity with microtemperatures

This paper is devoted to studying the linear system of partial differential equations modelling a one-dimensional thermo-porous-elastic problem with microtemperatures in the context of the dual-phase-lag heat conduction. Existence, uniqueness, and exponential decay of solutions are proved. Polynomial stability is also obtained in the case that the relaxation parameters satisfy a certain equality. Our arguments are based on the theory of semigroups of linear operators.

A generic construction for high order approximation schemes of semigroups using random grids

Our aim is to construct high order approximation schemes for general semigroups of linear operators \(P_{t},t\ge 0\). In order to do it, we fix a time horizon T and the discretization steps \(h_{l}=\frac{T}{n^{l}},l\in {\mathbb {N}}\) and we suppose that we have at hand some short time approximation operators \(Q_{l}\) such that \(P_{h_{l}}=Q_{l}+O(h_{l}^{1+\alpha })\) for some \(\alpha >0\). Then, we consider random time grids \(\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega )<\cdots <t_{m}(\omega )=T\}\) such that for all \(1\le k\le m\), \(t_{k}(\omega )-t_{k-1}(\omega )=h_{l_{k}}\) for some \(l_{k}\in {\mathbb {N}}\), and we associate the approximation discrete semigroup \(P_{T}^{\Pi (\omega )}=Q_{l_{n}}\ldots Q_{l_{1}}\). Our main result is the following: for any approximation order \(\nu \), we can construct random grids \(\Pi _{i}(\omega )\) and coefficients \(c_{i}\), with \(i=1,\ldots ,r\) such that $$\begin{aligned} P_{t}f=\sum _{i=1}^{r}c_{i}{\mathbb {E}}(P_{t}^{\Pi _{i}(\omega )}f(x))+O(n^{-\nu }) \end{aligned}$$with the expectation concerning the random grids \(\Pi _{i}(\omega ).\) Besides, \(Card (\Pi _{i}(\omega ))=O(n)\) and the complexity of the algorithm is of order n, for any order of approximation \(\nu \). The standard example concerns diffusion processes, using the Euler approximation for \(Q_l\). In this particular case and under suitable conditions, we are able to gather the terms in order to produce an estimator of \(P_tf\) with finite variance. However, an important feature of our approach is its universality in the sense that it works for every general semigroup \(P_{t}\) and approximations. Besides, approximation schemes sharing the same \(\alpha \) lead to the same random grids \(\Pi _{i}\) and coefficients \(c_{i}\). Numerical illustrations are given for ordinary differential equations, piecewise deterministic Markov processes and diffusions.

Spectrum and convergence of eventually positive operator semigroups

An intriguing feature of positive C_0-semigroups on function spaces (or more generally on Banach lattices) is that their long-time behaviour is much easier to describe than it is for general semigroups. In particular, the convergence of semigroup operators (strongly or in the operator norm) as time tends to infinity can be characterized by a set of simple spectral and compactness conditions. In the present paper, we show that similar theorems remain true for the larger class of (uniformly) eventually positive semigroups—which recently arose in the study of various concrete differential equations. A major step in one of our characterizations is to show a version of the famous Niiro–Sawashima theorem for eventually positive operators. Several proofs for positive operators and semigroups do not work in our setting any longer, necessitating different arguments and giving our approach a distinct flavour.

A Strong Convergence to a Common Fixed Point of a Subfamily of a Nonexpansive Evolution Family of Bounded Linear Operators on a Hilbert Space

In this article, we establish some results for convergence in a strong sense to a common fixed point of a subfamily of a nonexpansive evolution family of bounded linear operators on a Hilbert space. The obtained results generalize some existing ones in the literature for semigroups of operators. An example and an open problem are also given at the end.