Search

In this paper, we present a lemma that allows us to characterize a broad class of C0-semigroups on time scales, which can be applied to prove existence and uniqueness of solutions of systems of partial differential equations where the time domain is a time scale. The result obtained is applied to study the controllability of the heat equation on time scales. The lemma presented in this paper can be seen as a unification of the one proved by H. Leiva in [28], for semigroups in [0,∞). [ABSTRACT FROM AUTHOR]

Input-to-state stability estimates with respect to small initial conditions and input functions for infinite-dimensional systems with bilinear feedback are shown. We apply the obtained results to controlled versions of a viscous Burger equation with Dirichlet boundary conditions, a Schrödinger equation, a Navier--Stokes system, and a semilinear wave equation. [ABSTRACT FROM AUTHOR]

This paper considers a system with one robot and n safety units (one of which works while the others remain on standby), which is described by an integro-deferential equation. The system can fail in the following three ways: fails with an incident, fails safely and fails due to the malfunction of the robot. Using the C0–semigroups theory of linear operators, we first show that the system has a unique non-negative, time-dependent solution. Then, we obtain the exponential convergence of the time-dependent solution to its steady-state solution. In addition, we study the asymptotic behavior of some time-dependent reliability indices and present a numerical example demonstrating the effects of different parameters on the system.

In this paper, we prove the existance of subspace-diskcyclic C0-semigroups on any infinite-dimensional separable Banach space. We state that diskcyclic C0-semigroups are subspace-diskcyclic. Also, we establish some criteria for subspace-diskcyclic C0-semigroups. Most of these criteria are based on non-empty relatively open sets and some of them are based on dense sets. [ABSTRACT FROM AUTHOR]

In this paper, we consider optimal control of infinite dimensional bilinear systems in both cases unbounded and bounded controls set. Then, we minimise a functional cost constituted of the deviation between the desired state and the final one at time T, the effort term and the energy one. The purpose of this study is to prove that an optimal control exists and characterised as a solution to an optimality system. Thus we give a sufficient condition for the uniqueness of such a control. The obtained results are applied to many particular controls sets. Numerical algorithm for the computation of an optimal control is given and successfully illustrated through simulations for the heat and transport equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence and uniqueness of mild solutions to a semilinear Cauchy problem for an abstract fractional differential equation with state dependent nonlocal condition. Continuous dependence of solutions on initial conditions and local ...-approximate mild solution of the considered problem will be discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a discrete-time fractional model of abstract form involving the Riemann-Liouville-like difference operator. On account of the C0-semigroups generated by a closed linear operator A and based on a distinguished class of sequences of operators, we show the existence of stable solutions for the nonlinear Cauchy problem by means of fixed point technique and the compact method. Moreover, we also establish the Ulam-Hyers-Rassias stability of the proposed problem. Two examples are presented to explain the main results. [ABSTRACT FROM AUTHOR]

We study the initial value problem (*)u(n+1)−u(n)=Au(n+1)+∑k=0n+1a(n+1−k)Au(k),n∈N0u(0)=x,where A is closed linear operator defined on a Banach space X, x belongs to the domain of A, and the kernel a is a particular discretization of an integrable kernel a∈L1(R+). Assuming that A generates a resolvent family, we find an explicit representation of the solution to the initial value problem (*) as well as for its inhomogeneous version, and then we study the stability of such solutions. We also prove that for a special class of kernels a, it suffices to assume that A generates an immediately norm continuous C0‐semigroup. We employ a new computational method based on the Poisson transformation. [ABSTRACT FROM AUTHOR]

A mathematical model for the dynamics of prion proliferation in the presence of chaperone involving a coupled system consisting of an ordinary differential equation and a partial integro‐differential equation is analyzed. For bounded reaction rates, we prove the existence and uniqueness of positive classical solutions with the help of the theory of evolution system. In the case of unbounded reaction rates, the model is set up into a semilinear evolution equation form in the product Banach space ℝ×L1(z0,∞);(q+z)dz and the existence of a unique positive local mild solution is established by using C0‐semigroups theory of operators. [ABSTRACT FROM AUTHOR]

In this paper, we prove the global in time solvability of the continuous growth--fragmentation--coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth--fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic. This article is part of the theme issue 'Semigroup applications everywhere'. [ABSTRACT FROM AUTHOR]

Let X,Y be Banach spaces, (St)t≥0 a C0-semigroup on X, −A the corresponding infinitesimal generator on X, C a bounded linear operator from X to Y, and T>0. We consider the system x˙(t)=−Ax(t),y(t)=Cx(t)t∈(0,T],x(0)=x0∈X. We provide sufficient conditions such that this system satisfies a final state observability estimate in Lr((0,T);Y), r∈[1,∞]. These sufficient conditions are given by an uncertainty relation and a dissipation estimate. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider the example where A is an elliptic operator in Lp(Rd) for 1

During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C0-semigroups of linear and continuous operators.

We study the rate of energy decay for solutions of a coupled wave-heat system on a rectangular domain. Using techniques from the theory of C0-semigroups, and in particular a well-known result due to Borichev and Tomilov, we prove that the energy of classical solutions decays like t-2/3 as t → ∞. This rate is moreover shown to be sharp. Our result implies in particular that a general estimate in the literature, which predicts at least logarithmic decay and is known to be best possible in general, is suboptimal in the special case under consideration here. Our strategy of proof involves direct estimates based on separation of variables and a refined version of the technique developed in our earlier paper for a one-dimensional wave-heat system. [ABSTRACT FROM AUTHOR]

We consider the well-posedness of a class of hyperbolic partial differential equations on a one-dimensional spatial domain. This class includes in particular infinite networks of transport, wave and beam equations, or even combinations of these. Equivalent conditions for contraction semigroup generation are derived. We consider these equations on a finite interval as well as on a semi-axis. [ABSTRACT FROM AUTHOR]

For nonuniform exponentially bounded evolution families on the half-line, we introduce a special class of Banach function spaces, on which we define certain C0-semigroups. We characterize the existence of nonuniform exponential stability in terms of invertibility of the corresponding infinitesimal generators. The invertibility of these generators is connected to a particular type of admissible exponents that are specific to nonuniform behavior. For the bounded orbits, nonuniform exponential stability results from a spectral property of generators. The C0-semigroups we deal with verify the spectral mapping theorem, as well as the evolution semigroups, in the uniform case. In particular, our results directly apply to all linear differential equations with finite Lyapunov exponent. [ABSTRACT FROM AUTHOR]

The Ulam stability of the composition of two Ulam stable operators has been investigated by several authors. Composition of operators is a key concept when speaking about C0-semigroups. Examples of C0-semigroups formed with Ulam stable operators are known. In this paper, we construct a C0-semigroup (Rt)t≥0 on C[0,1] such that for each t>0, Rt is Ulam unstable. Moreover, we compute the central moments of Rt and establish a Voronovskaja-type formula. This enables to prove that C2[0,1] is contained in the domain D(A) of the infinitesimal generator of the semigroup. We raise the problem to fully characterize the domain D(A).

Working in the setting of Banach spaces, we give a simpler proof of a result concerning the Ulam stability of the composition of operators. Several applications are provided. Then, we give an example of a discrete semigroup with Ulam unstable members and an example of Ulam stable operators on a Banach space, such that their sum is not Ulam stable. Another example is concerned with a C 0 -semigroup ( T t ) t ≥ 0 of operators for which each T t is Ulam stable. We present an open problem concerning the Ulam stability of the members of the Bernstein C 0 -semigroup. Two other possible problems are mentioned at the end of the paper.

We extend to the context of Lp spaces and C0-semigroups of operators our previous results from Heilmann and Raşa (Positivity 21:897-910, 2017. 10.1007/s11117-016-0441-1), concerning the eigenstructure and iterates of uniquely ergodic Kantorovich modifications of linking operators. [ABSTRACT FROM AUTHOR]

This paper investigates the asymptotic behaviour of solutions to certain infinite systems of coupled recurrence relations. In particular, we obtain a characterisation of those initial values which lead to a convergent solution, and for initial values satisfying a slightly stronger condition we obtain an optimal estimate on the rate of convergence. By establishing a connection with a related problem in continuous time, we are able to use this optimal estimate to improve the rate of convergence in the continuous setting obtained by the authors in a previous paper. We illustrate the power of the general approach by using it to study several concrete examples, both in continuous and in discrete time. [ABSTRACT FROM AUTHOR]

It is known that a differential equation ẋ(t) = Ax(t) on the Banach space X (we assume the well-posedness, i.e., A generates a C0-semigroup {T(t)}t ≥ 0) is hyperbolic if X can be decomposed as X = X1 ⊕ X2 so that solutions ẋ(∙) starting from X1 (respectively, from X2) decay exponentially in forward time (respectively, in backward time). Hyperbolicity forces the solutions that start from X2 to exist for negative time (or, equivalently, the semigroup generated by A to extend to a C0-group on X2). We generalize this notion by replacing the exponential decay in negative time for the solutions starting in X2 with an exponential blow-up in positive time (we call this an exponential dichotomy). It is obvious that hyperbolicity implies the existence of an exponential dichotomy, but the converse is not valid (we point out an example in this context). We obtain a unified treatment of admissibility-type conditions guaranteeing the existence of an exponential dichotomy and complete characterizations of the hyperbolicity of autonomous differential equations. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with distributional chaos and the principal measure of C 0 -semigroups on a Frechet space. New definitions of strong irregular (semi-irregular) vectors are given. It is proved that if C 0 -semigroup T has strong irregular vectors, then T is distributional chaos in a sequence, and the principal measure μ p ( T ) is 1. Moreover, T is distributional chaos equivalent to that operator T t   is distributional chaos for every ∀ t > 0 .

We consider the C0-semigroups generated on the spaces of continuous functions, respectively, square-integrable functions, by the differential operator The relationship between these semigroups as well as several qualitative properties of them is studied. We discuss the possibility of approximating the semigroups in terms of suitable iterations of positive linear operators. [ABSTRACT FROM AUTHOR]

In this work, we study the controllability for a class of nonlinear integrod-ifferential equations in Banach spaces. Using resolvent operator properties and Schaefer's fixed point Theorem, we give sufficient conditions for the controllability when the nonlinear term satisfies the Carathéodory conditions. [ABSTRACT FROM AUTHOR]

Consider the classical solutions of the abstract approximate problems *** where An generates a sequence of C0-semigroups of operators Tn(t) on the Hilbert spaces Hn. Classical solutions of this problem may converge to 0 polynomially, but not exponentially, in the following sense *** for some constants Cn, á and â > 0. This paper has two objectives. First, necessary and sufficient conditions are given to characterize the uniform polynomial stability of the sequence Tn(t) on Hilbert spaces Hn. Secondly, approximation in control of a one-dimensional hyperbolic-parabolic coupled system subject to Dirichlet. Dirichlet boundary conditions, is considered. The uniform polynomial stability of corresponding semigroups associated with approximation schemes is proved. Numerical experimental results are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the abstract Cauchy problem for an elliptic equation. This problem is well known as severely ill-posed. The goal of this paper is to present some extensions of the quasi-reversibility method applied to an ill-posed Cauchy problem for elliptic equations. The key point to our analysis is the use of the general modified quasi-reversibility method to construct a family of regularizing operators for the considered problem and we prove the convergence of this method. [ABSTRACT FROM AUTHOR]