Semigroups Of Operators Research Articles

Functional equations in formal power series

Let \(k\) be an algebraically closed field of characteristic zero, and \(k[[z]]\) the ring of formal power series over \(k\). In this paper, we study equations in the semigroup \(z^2k[[z]]\) with the semigroup operation being composition. We prove a number of general results about such equations and provide some applications. In particular, we answer a question of Horwitz and Rubel about decompositions of "even" formal power series. We also show that every right amenable subsemigroup of \(z^2k[[z]]\) is conjugate to a subsemigroup of the semigroup of monomials.

Upper and lower estimates for rate of convergence in the Chernoff product formula for semigroups of operators

Upper and lower estimates for rate of convergence in the Chernoff product formula for semigroups of operators

Decomposition of pseudo-uninorms with continuous underlying functions via ordinal sum

The decomposition of all pseudo-uninorms with continuous underlying functions, defined on the unit interval, via Clifford's ordinal sum is described. It is shown that each such pseudo-uninorm can be decomposed into representable and trivial semigroups, and special semigroups defined on two points, where the corresponding semigroup operation is the projection to one of the coordinates. Linear orders, for which the ordinal sum of such semigroups yields a pseudo-uninorm, are also characterized.

Adaptive vibration control for stabilisation of the Euler–Bernoulli beam with an unknown payload

In this paper, an adaptive boundary controller is designed to solve the boundary control problem of Euler–Bernoulli beams with an unknown payload. Therefore, a boundary controller with an adaptive law is designed to compensate for the parameter uncertainty of the system. Partial Differential Equations (PDEs) are used to describe the Euler–Bernoulli beam system. The well-posedness of the system under the action of an adaptive boundary controller is proved by using the linear operator semigroup method. Meanwhile, the asymptotic stability of the closed-loop system is derived from the extended Krasovskii–LaSalle invariance principle. The effectiveness and superiority of the proposed method are illustrated by simulation in comparison with the existing results.

On stability of iteration processes convergent to stationary points of semigroups of nonlinear operators in metric spaces

Let C be a closed subset of a complete metric space. In this paper we study the stability problem for some iteration processes that approximate stationary points of pointwise Lipschitzian semigroups of nonlinear mappings acting within C. We show that a large class of well-controlled processes is stable under summable errors. These results are then applied to the stability of some known processes including the generalized Krasnosel'skii-Mann and the two-step Ishikawa iteration processes.

On sharp heat kernel estimates in the context of Fourier–Dini expansions

We prove sharp estimates of the heat kernel associated with Fourier–Dini expansions on (0,1) equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result including sharp bounds for the corresponding Poisson and potential kernels, sharp mapping properties of the maximal heat semigroup and potential operators and boundary convergence of the Fourier–Dini semigroup.

Local and nonlocal problems for fractional impulsive evolution systems with order $\nu\in(1,2)$

In this paper, we study the existence and uniqueness of $PC$-mild solutions for fractional impulsive evolution systems with Caputo derivative. First, we give a compact result of the solution operators of linear fractional evolution system while the cosine family is compact. Second, the representation of $PC$-mild solutions of linear fractional impulsive systems with initial value conditions and nonlocal initial value conditions are given by using the Laplace transform method. Third, several sufficient conditions for judging the existence and uniqueness of solutions of our problem are established via some fixed point theorems and operator semigroup theory. Finally, an example is given to illustrate the results of our paper.

Discussion on the existence and controllability of Hilfer fractional stochastic differential equation with non-dense domain

This article studies the approximate controllability for a class of fractional control system with semigroup operators governed by differential equations with Hilfer fractional derivatives of order α ∈ ( 0 , 1 ) and type β ∈ [ 0 , 1 ] in Hilbert spaces. First, we establish a set of sufficient conditions for the approximate controllability for Hilfer fractional stochastic differential equation with non-dense domain in Hilbert spaces. The existence results are obtained by using the fractional calculus, semi-group theory, Wiener process and fixed point technique to prove our main results. Further, we extend the result to study the approximate controllability concept. An example is provided to illustrate the application of the obtained theory.

Inertial Invariant Manifolds of a Nonlinear Semigroup of Operators in a Hilbert Space

Inertial Invariant Manifolds of a Nonlinear Semigroup of Operators in a Hilbert Space

Some Families of Random Fields Related to Multiparameter Lévy Processes

Let R+N=[0,∞)N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^N_+= [0,\\infty )^N$$\\end{document}. We here make new contributions concerning a class of random fields (Xt)t∈R+N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X_t)_{t\\in {\\mathbb {R}}^N_+}$$\\end{document} which are known as multiparameter Lévy processes. Related multiparameter semigroups of operators and their generators are represented as pseudo-differential operators. We also provide a Phillips formula concerning the composition of (Xt)t∈R+N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X_t)_{t\\in {\\mathbb {R}}^N_+}$$\\end{document} by means of subordinator fields. We finally define the composition of (Xt)t∈R+N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X_t)_{t\\in {\\mathbb {R}}^N_+}$$\\end{document} by means of the so-called inverse random fields, which gives rise to interesting long-range dependence properties. As a byproduct of our analysis, we present a model of anomalous diffusion in an anisotropic medium which extends the one treated in Beghin et al. (Stoch Proc Appl 130:6364–6387, 2020), by improving some of its shortcomings.

Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability

AbstractWe prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from $$H^\alpha ({\mathbb {R}},U)$$ H α ( R , U ) to $$H^\beta ({\mathbb {R}},U)$$ H β ( R , U ) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.

A study on approximate controllability of linear impulsive equations in Hilbert spaces

In this paper, we study an approximate controllability for the impulsive linear evolution equations in Hilbert spaces. We give a representation of solution in terms of semigroup and impulsive operators. We present the necessary and sufficient conditions for approximate controllability of linear impulsive evolution equation in terms of impulsive resolvent operator. An example is provided to illustrate the application of the obtained results.

Збурення ізотропного $\alpha$-стійкого випадкового процесу псевдоградієнтом з узагальненим коефіцієнтом

The article is devoted to the perturbation of an isotropic $\alpha$-stable stochastic process in a finite-dimensional Euclidean space by a pseudo-gradient operator multiplied by a delta-function on a hypersurface. This is analogous to the construction of some membrane in the phase space. Semigroup of operators on the space of continuous bounded functions is constructed. It has the infinitesimal generator (in some generalized sense) $c\Delta_\alpha+(q\delta_S\nu,\nabla_\beta)$, where $c$ is some positive constant, $\Delta_\alpha$ is the fractional Laplacian of the order $\alpha$, $\delta_S$ is the delta-function on the hypersurface $S$, which has a normal vector $\nu$, $q$ is some continuous bounded function, $\nabla_\beta$ is a fractional gradient (pseudo-gradient), that is the pseudo-differentional operator defined by the symbol $i\lambda|\lambda|^{\beta-1}$. The order of the pseudo-gradient is less than $\alpha-1$. Some properties of the obtained semigroup are investigated. This semigroup defines a pseudo-process.

The non-homogeneous Cauchy problem of semigroup of linear operators

This paper presents the results of ω-order reversing partial contraction mapping generated by the non-homogeneous Cauchy problem. We investigated several concepts of solution for non-homogeneous Cauchy problem where an analysis was arrived out on the relationship between C' strong and respectively C0-solution. We showed that Z is the infinitesimal generator of a C0-semigroup of contractions. Finally, we obtained that the problem has a unique classical solution and f, a member of L1(a,b; X) is continuous on (a,b).

Robust vibration control for a flexible cable gantry crane system subject to boundary disturbance

This paper addresses the robust vibration control problem for a gantry crane system which is governed by a partial differential equation. The control objectives are to transport the cargo from the initial position to the desired position and suppress the vibration of the cable and payload in the presence of boundary disturbance. To achieve the objectives, a sliding-mode controller is proposed by repressing the boundary disturbance. By using the operator semigroup theory, the well-posedness of the closed-loop system is guaranteed. The asymptotic stability of the closed-loop system is proven by using the extended LaSalle’s invariance principle. Both numerical simulations and experiments are provided to illustrate the effectiveness of the proposed boundary control method.

Decay of operator semigroups, infinite-time admissibility, and related resolvent estimates

We study decay rates for bounded C0-semigroups from the perspective of Lp-infinite-time admissibility and related resolvent estimates. In the Hilbert space setting, polynomial decay of semigroup orbits is characterized by the resolvent behavior in the open right half-plane. A similar characterization based on Lp-infinite-time admissibility is provided for multiplication semigroups on Lq-spaces with 1≤q≤p<∞. For polynomially stable C0-semigroups on Hilbert spaces, we also give a sufficient condition for L2-infinite-time admissibility.

Ideal categories of the normed algebra of finite rank bounded operators on a Hilbert space

In this paper, we introduce the normal category [Formula: see text] [[Formula: see text]] of principal left [right] ideals of the normed algebra [Formula: see text] of all finite rank bounded operators on a Hilbert space [Formula: see text] and show that [Formula: see text] and [Formula: see text] are isomorphic using Hilbert space duality. We also prove that the semigroup of all normal cones in [Formula: see text] is isomorphic to the semigroup of all finite rank operators on [Formula: see text]. Further, we construct bounded normal cones in [Formula: see text] such that the set of all bounded normal cones in [Formula: see text] forms a normed algebra isomorphic to the normed algebra [Formula: see text].

Connections of Green's relations of a Γ-semigroup with operator semigroups

The theory of Γ-semigroups is an extension of the semigroup theory. In this paper, we examine the left operator semigroup L and the right operator semigroup R via modified definition of Γ-semigroup and deduce some results of operator semigroups acting on a Γ-semigroup. Further, we study some relationships between Green’s equivalence relations of a Γ-semigroup and its left (right) operator semigroup. In particular, we show that if two elements of a Γ-semigroup S are L(R)-related, then the two elements of L(R) resulting from S for every α ∈ Γ are also L(R)-related. Also, we describe that if two elements of S are α and β-idempotent such that the two elements are R-related in L, then their R-relation holds in S for some α, β ∈ Γ.

Rational approximation of operator semigroups via the [formula omitted]-calculus

We improve the classical results by Brenner and Thomée on rational approximations of operator semigroups. In the setting of Hilbert spaces, we introduce a finer regularity scale for initial data, provide sharper stability estimates, and obtain optimal approximation rates. Moreover, we strengthen a result due to Egert-Rozendaal on subdiagonal Padé approximations of operator semigroups. Our approach is direct and based on the theory of the B- functional calculus developed recently. On the way, we elaborate a new and simple approach to construction of the B-calculus thus making the paper essentially self-contained.

Iterative learning control of an Euler-Bernoulli beam with time-varying boundary disturbance

This paper investigates the stabilization of Euler-Bernoulli beam systems modeled by a partial differential equation (PDE) with unknown time-varying disturbance. An iterative learning controller is designed using only boundary state feedback to realize the vibration control subject to unknown boundary disturbance. The well-posedness for the closed-loop system is given by the operator semigroup theory. Furthermore, the exponentially stable for the closed-loop system is proved by the Lyapunov method. The comparisons with existing results are made to demonstrate the effectiveness and advantages of the proposed boundary iterative learning control method.