Linear Evolution Operator Research Articles

Approximate Controllability of Non-autonomous Second Order Impulsive Functional Evolution Equations in Banach Spaces

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder’s fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we provide two concrete examples to validate our results.

Time-periodic Gelfand-Shilov spaces and global hypoellipticity on [formula omitted

We introduce a class of time-periodic Gelfand-Shilov spaces of functions on T×Rn, where T∼R/2πZ is the one-dimensional torus. We develop a Fourier analysis inspired by the characterization of the Gelfand-Shilov spaces in terms of the eigenfunction expansions given by a fixed normal, globally elliptic differential operator on Rn. In this setting, as an application, we characterize the global hypoellipticity for a class of linear differential evolution operators on T×Rn.

On the Spectral Properties for the Linearized Problem around Space-Time-Periodic States of the Compressible Navier–Stokes Equations

This paper studies the linearized problem for the compressible Navier-Stokes equation around space-time periodic state in an infinite layer of Rn (n=2,3), and the spectral properties of the linearized evolution operator is investigated. It is shown that if the Reynolds and Mach numbers are sufficiently small, then the asymptotic expansions of the Floquet exponents near the imaginary axis for the Bloch transformed linearized problem are obtained for small Bloch parameters, which would give the asymptotic leading part of the linearized solution operator as t→∞.

Linear Stochastic Differential Equations Driven by Gauss-Volterra Processes and Related Linear-Quadratic Control Problems

A stochastic linear-quadratic control problem is formulated and solved for some stochastic equations in an infinite dimensional Hilbert space for both finite and infinite time horizons. The equations are bilinear in the state and the noise process where the noise is a scalar Gauss-Volterra process. The Gauss-Volterra noise processes are obtained from the integral of a Brownian motion with a suitable kernel function. These noise processes include fractional Brownian motions with the Hurst parameter $$H \in (\frac{1}{2},1)$$ , Liouville fractional Brownian motions with $$H \in (\frac{1}{2},1)$$ , and some multifractional Brownian motions. The family of admissible controls for the quadratic costs is a family of linear feedback controls. This restriction on the family of controls allows for a feasible implementation of the optimal controls. The bilinear equations have drift terms that are linear evolution operators. These equations can model stochastic partial differential equations of parabolic and hyperbolic types and two families of examples are given.

Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay

The controllability of non-autonomous evolution systems is an important and difficult topic in control theory. In this paper, we study the approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. The theory of linear evolution operators is used instead of \begin{document}$C_0-$\end{document} semigroup to discuss the problem. Some sufficient conditions of approximate controllability are formulated and proved here by using the resolvent operator condition. Finally, two examples are provided to illustrate the applications of the obtained results.

Actuator selection and placement for localized feedback flow control

The selection and placement of actuators and sensors to control compressible viscous flows is addressed by developing a novel methodology based upon the eigensystem structural sensitivity of the linearized evolution operator appropriate for linear feedback control. Forward and adjoint global modes are used to construct a space of possible perturbations to the linearized operator, which yields a small optimization problem for selecting the parameters that best achieve the control objective, including where they should be placed. The method is demonstrated by informing actuation to suppress amplification of the instabilities in boundary layer separation in a high-subsonic diffuser. Complete stabilization is observed in the separated shear layer for short downstream distances at modest Reynolds number. Higher Reynolds numbers and longer distances are expected to be more challenging to stabilize; here the control informed by the procedure still substantively suppresses amplification of instabilities. It is also demonstrated that more complex actuator–sensor selections may not yield superior controllers.

Semilinear inclusions with nonlocal conditions without compactness in non-reflexive spaces

An existence result for an abstract nonlocal boundary value problem $x'\in A(t)x(t)+F(t,x(t))$, $Lx\in B(x)$, is given, where $A(t)$ determines a linear evolution operator, $L$ is linear, and $F$ and $B$ are multivalued. To avoid compactness conditions, the weak topology is employed. The result applies also in nonreflexive spaces under a hypothesis concerning the De Blasi measure of noncompactness. Even in the case of initial value problems, the required condition is essentially milder than previously known results.

On linear and nonlinear aspects of dynamic mode decomposition

SummaryThe approximation of reduced linear evolution operator (propagator) via dynamic mode decomposition (DMD) is addressed for both linear and nonlinear events. The 2D unsteady supersonic underexpanded jet, impinging the flat plate in nonlinear oscillating mode, is used as the first test problem for both modes. Large memory savings for the propagator approximation are demonstrated. Corresponding prospects for the estimation of receptivity and singular vectors are discussed. The shallow water equations are used as the second large‐scale test problem. Excellent results are obtained for the proposed optimized DMD method of the shallow water equations when compared with recent POD‐based/discrete empirical interpolation‐based model reduction results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.

Spectral tests for observability and detectability of periodic Markov jump systems with nonhomogeneous Markov chain

This paper addresses observability and detectability of discrete-time periodic systems with nonhomogeneous Markov jump parameter. Popov–Belevitch–Hautus (PBH)-type criteria are proposed for the concerned structural properties. It is shown that, different from stochastic systems with constant coefficients or homogeneous Markov chain, the spectral criteria of considered plants do not rely on a single operator but on a finite sequence of linear evolution operators. By use of the obtained detectability criterion, an extended Lyapunov theorem is established, which relates asymptotic mean square stability to a periodic Lyapunov equation. Further, a difference Riccati equation with periodic coefficients is studied and some sufficient conditions are presented for the existence of stabilizing solution.

Stationary solutions of Keller–Segel-type crowd motion and herding models : Multiplicity and dynamical stability

In this paper we study two models for crowd motion and herding. Each of the models is of Keller-Segel type and involves two parabolic equations, one for the evolution of the density and one for the evolution of a mean field potential. We classify all radial stationary solutions, prove multiplicity results and establish some qualitative properties of these solutions, which are characterized as critical points of an energy functional. A notion of variational stability is associated to such solutions. The dynamical stability in a neighborhood of a stationary solution is also studied in terms of the spectral properties of the linearized evolution operator. For one of the two models, we exhibit a Lyapunov functional which allows to make the link between the two notions of stability. Even in that case, for certain values of the mass parameter and all other parameters taken in an appropriate range, we find that two dynamically stable stationary solutions exist. We further discuss qualitative properties of the solutions using theoretical methods and numerical computations.

Nonuniform dichotomy spectrum and normal forms for nonautonomous differential systems

The aim of this paper is to study the normal forms of nonautonomous differential systems. For doing so, we first investigate the nonuniform dichotomy spectrum of the linear evolution operators that admit a nonuniform exponential dichotomy, where the linear evolution operators are defined by nonautonomous differential equations x˙=A(t)x in Rn. Using the nonuniform dichotomy spectrum we obtain the normal forms of the nonautonomous linear differential equations. Finally we establish the finite jet normal forms of the nonlinear differential systems x˙=A(t)x+f(t,x) in Rn, which is based on the nonuniform dichotomy spectrum and the normal forms of the nonautonomous linear systems.

Dynamic stability of spinning viscoelastic cylinders at finite deformation

The study of spinning axisymmetric cylinders undergoing finite deformation is a classic problem in several industrial settings – the tire industry in particular. We present a stability analysis of spinning elastic and viscoelastic cylinders using ARPACK to compute eigenvalues and eigenfunctions of finite element discretizations of the linearized evolution operator. We show that the eigenmodes correspond to N-peak standing or traveling waves for the linearized problem with an additional index describing the number of oscillations in the radial direction. We find a second hierarchy of bifurcations to standing waves where these eigenvalues cross zero, and confirm numerically the existence of finite-amplitude standing waves for the nonlinear problem on one of the new branches. In the viscoelastic case, this analysis permits us to study the validity of two popular models of finite viscoelasticity. We show that a commonly used finite deformation linear convolution model results in non-physical energy growth and finite-time blow-up when the system is perturbed in a linearly unstable direction and followed nonlinearly in time. On the other hand, Sidoroff-style viscoelastic models are seen to be linearly and nonlinearly stable, as is physically required.

Asymptotic Estimates for the Parabolic-Elliptic Keller-Segel Model in the Plane

We investigate the large-time behavior of the solutions of the two-dimensional Keller-Segel system in self-similar variables, when the total mass is subcritical, that is less than 8 π after a proper adimensionalization. It was known from previous works that all solutions converge to stationary solutions, with exponential rate when the mass is small. Here we remove this restriction and show that the rate of convergence measured in relative entropy is exponential for any mass in the subcritical range, and independent of the mass. The proof relies on symmetrization techniques, which are adapted from a paper of Diaz et al. and allow us to establish uniform estimates for L p norms of the solution. Exponential convergence is obtained by the mean of a linearization in a space which is defined consistently with relative entropy estimates and in which the linearized evolution operator is self-adjoint. The core of proof relies on several new spectral gap estimates which are of independent interest.

Linear Instability Criteria for Ideal Fluid Flows Subject to Two Subclasses of Perturbations

We examine the linear stability of both 2D and 3D steady state solutions to Euler’s equation subject to two classes of high frequency perturbations: those that preserve the topology of vortex lines and those that do not. Lower bounds for the essential spectral radius of the linear evolution operator are given for both types of perturbations.

Continuous spectra of the Batchelor vortex

The spectra of the Batchelor vortex are obtained by discretizing its linearized evolution operator using a modified Chebyshev polynomial approximation at a Reynolds number of 1000 and zero azimuthal wavenumber. Three types of eigenmodes are identified from the spectra: discrete modes, potential modes and free-stream modes. The discrete modes have been extensively documented but the last two modes have received little attention. A convergence study of the spectra and pseudospectra supports the classification that discrete modes correspond to discrete spectra while the other two modes correspond to continuous spectra. Free-stream modes have finite amplitude in the far field whilst potential modes decay to zero in the far field. The free-stream modes are therefore a limiting form of the potential modes when the radial decay rate of velocity components reduces to zero. The radial form of the free-stream modes with axial and radial wavenumbers is investigated and the penetration of the free-stream mode into the vortex core highlights the possibility of interaction between the potential region and the vortex core. A wavepacket pseudomode study confirms the existence of continuous spectra and predicts the locations and radial wavenumbers of the eigenmodes. The pseudomodes corresponding to the potential modes are observed to be in the form of one or two wavepackets while the free-stream modes are not observed to be in the form of wavepackets.

Spectral properties of chimera states

Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a spatiotemporal pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that, in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.

Local existence of solutions for the Neumann problem of the nonlinear elastodynamic system outside a domain

In this paper, we study the mixed initial–boundary value problem of Neumann type for the nonlinear elastic wave equation outside a domain. The local existence of solutions to this problem is proved by iteration. To get this result, we prove the existence of solutions for the second order linear hyperbolic system with variable coefficients (in Sobolev spaces) outside of a domain by using linear evolution operators and integro-differential equations.

Polynomial stability of evolution operators in Banach spaces

The paper considers three concepts of polynomial stability for linear evolution operators which are defined in a general Banach space and whose norms can increase not faster than exponentially. Our approach is based on the extension of techniques for exponential stability to the case of polynomial stability. Some illustrating examples clarify the relations between the stability concepts considered in paper. The obtained results are generalizations of well-known theorems about the uniform and nonuniform exponential stability.

Converse Lyapunov Theorems for Switched Systems in Banach and Hilbert Spaces

We consider switched systems on Banach and Hilbert spaces governed by strongly continuous one-parameter semigroups of linear evolution operators. We provide necessary and sufficient conditions for their global exponential stability, uniform with respect to the switching signal, in terms of the existence of a Lyapunov function common to all modes.

Linear study of the nonmodal growth of drift waves in dusty plasmas

The main effect of dust on drift wave turbulence is the enhancement of the nonadiabaticity. Previous work found that nonmodal behavior is important in the nonadiabatic regime of the drift wave system. Here, the modal and nonmodal properties of the linear Hasegawa–Wakatani system of dusty plasmas are investigated. The non-normality of the linear evolution operator can lead to enhanced transient growth rates compared to the modal growth rates.