Systems In Finite-dimensional Spaces Research Articles

Existence of Solutions and Relative Controllability of a Stochastic System with Nonpermutable Matrix Coefficients

In this study, time-delayed stochastic dynamical systems of linear and nonlinear equations are discussed. The existence and uniqueness of the stochastic semilinear time-delay system in finite dimensional space is investigated. Introducing the delay Gramian matrix, we establish some sufficient and necessary conditions for the relative approximate controllability of time-delayed linear stochastic dynamical systems. In addition, by applying the Banach fixed point theorem, we establish some sufficient relative approximate controllability conditions for semilinear time-delayed stochastic differential systems. Finally, concrete examples are given to illustrate the main results.

Synchronization of Fractional Stochastic Chaotic Systems via Mittag-Leffler Function

This paper is involved with synchronization of fractional order stochastic systems in finite dimensional space, and we have tested its time response and stochastic chaotic behaviors. Firstly, we give a representation of solution for a stochastic fractional order chaotic system. Secondly, some useful sufficient conditions are investigated by using matrix type Mittag-Leffler function, Jacobian matrix via stochastic process, stability analysis and feedback control technique to assure the synchronization of stochastic error system. Thereafter, numerical illustrations are provided to verify the theoretical parts.

Relative controllability of impulsive multi-delay differential systems

In this paper, relative controllability of impulsive multi-delay differential systems in finite dimensional space are studied. By introducing the impulsive multi-delay Gramian matrix, a necessary and sufficient condition, and the Gramian criteria, for the relative controllability of linear systems is given. Using Krasnoselskii’s fixed point theorem, a sufficient condition for controllability of semilinear systems is obtained. Numerically examples are given to illustrate our theoretically results.

Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in infinite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in finite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in infinite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is introduced. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research.

Stabilization on Periodic Impulse Control Systems

This paper studies the stabilization for a kind of linear and impulse control system in finite-dimensional space where impulse instants appear periodically. We present several characterizations on the stabilization; show how to design feedback laws, and provide locations for impulse instants to ensure the stabilization. In the proofs of these results, we set up a discrete LQ problem, derive a discrete dynamic programming principle, build up a variant of Riccati's equation, apply repeatedly the Kalman controllability decomposition, and use a controllability result built up in [S. Qin and G. Wang, J. Differential Equations, 263 (2017), pp. 6456--6493].

Null controllability results for stochastic delay systems with delayed perturbation of matrices

Abstract In this paper, we adopt a notation of delayed perturbation of matrices to study controllability of stochastic delay systems in finite dimensional spaces. By using the delayed perturbation of matrix, we write an explicit formula of solutions to linear stochastic delay systems. We present the necessary and sufficient conditions for null controllability of linear stochastic delay systems by using perturbed controllability matrix and rank correlation method. Thereafter, sufficient conditions are derived for null controllability of semilinear stochastic delay systems under the proved results of corresponding linear stochastic system is null controllable, stochastic linear controllability operator, appropriate hypothesis on semilinear term and stochastic delay analysis techniques. The obtained theoretical results is verified through numerical simulation.

On the Controllability of Conformable Fractional Deterministic Control Systems in Finite Dimensional Spaces

In this paper, we establish a set of convenient conditions of controllability for semilinear fractional finite dimensional control systems involving conformable fractional derivative. Indeed, sufficient conditions of controllability for a semilinear conformable fractional system are presented, assuming that the corresponding linear systems are controllable. The present method is based on conformable fractional exponential matrix, Gramian matrix, and the iterative technique. Two illustrated examples are carried out to establish the facility and efficiency of this technique.

Description of the information process as a discrete stream.

The work considers the forecasting of the process of functioning of a dynamic complex system, which is a big problem because of the considerable volume of information circulating in it and many random factors affecting its operation. In this paper, the structure of the information process described in the form of a discrete information flow is considered. To determine portions of the information flow information for the formation of a control action, these flows are identified and structured based on determining the dimension of the attractors of dynamical systems in finite-dimensional spaces. To manage a discrete information process, the number of control checks that make it possible to determine the duration of this process predicts it. In this case, the pieces of information and time intervals for each portion of the discrete information stream are considered the same. As a result of the conducted researches the program complex on formation of predicted parameters of training with definition of portions of training information is developed. The paper gives comparative results of the duration of training - experimental and predicted.

Results on Controllability of Nonlinear Hilfer Fractional Stochastic System

Abstract The main objective of this paper is to present sufficient conditions for controllability of Hilfer fractional nonlinear stochastic systems in finite dimensional space. The main results are obtained by using the Nussbaum fixed point theorem, stochastic analysis approach and generalized fractional calculus (Hilfer fractional derivative) which is universality of Riemann-Liouville and Caputo fractional derivative. Finally, a numerical example is provided to show the effectiveness of the obtained theoretical result. The obtained result is more generalized one than the existing results on fractional stochastic system in finite dimensional space.

Asymptotic analysis of the parametric instability of nonlinear hyperbolic equations

This paper is concerned with parametric resonance under nonlinear periodic perturbations of differential equations which are abstract analogues of hyperbolic systems. A modification of the Krylov-Bogolyubov averaging method capable of circumventing the well-known small divisor problem is applied to reduce the description of solutions of perturbed equations at resonance to the study of autonomous dynamical systems in finite-dimensional spaces. Bibliography: 28 titles.

The Controllability of Fractional Damped Stochastic Integrodifferential Systems

AbstractIn this paper, the complete controllability is investigated for nonlinear fractional damped stochastic integrodifferential system in finite dimensional space. Using linear controllability theory, sufficient conditions ensuring complete controllability are derived based on controllability Grammian matrix which is defined by Mittag‐Leffler matrix function and fixed‐point techniques. Finally, two numerical examples are given to verify the proposed conditions.

On controllability for a class of second order semilinear stochastic systems

Many dynamical systems are modeled as second order differential systems. In this paper we address the problem of controllability for second order semilinear stochastic systems in finite dimensional spaces. Sufficient criteria for the complete controllability are formulated under some appropriate assumptions. The results are obtained by using the Banach fixed point theorem. Finally an illustrative example is provided.

Existence and uniqueness of periodic solutions for gradient systems in finite dimensional spaces

This paper deals with an abstract periodic gradient system in which the gradient is taken with respect to a variable metric. We obtain an existence and uniqueness result via the application of a global inverse theorem.

Observability and Controllability of Fractional Linear Dynamical Systems

In this paper we study the observability and controllability of fractional linear dynamical systems in finite dimensional spaces. Examples are included to illustrate the theoretical results proved in this manuscript.

On the controllability of fractional dynamical systems

Abstract This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.

On the stability of abstract monotone impulsive differential equations in terms of two measures

We consider differential equations in a Banach space subjected to an impulsive influence at fixed times. It is assumed that a partial ordering is introduced in the Banach space by using a normal cone and that the differential equations are monotone with respect to the initial data. We propose a new approach to the construction of comparison systems in finite-dimensional spaces without using auxiliary Lyapunov-type functions. On the basis of this approach, we establish sufficient conditions for the stability of this class of differential equations in terms of two measures. In this case, a Birkhoff measure is chosen as the measure of initial displacements, and the norm in the given Banach space is used as the measure of current displacements. We present some examples of investigations of the impulsive systems of differential equations in the critical cases and linear impulsive systems of partial differential equations.

Controllability Results for Second Order Neutral Impulsive Integrodifferential Systems

In this paper, we address the problem of controllability for abstract second order neutral integrodifferential systems with impulsive conditions in Banach spaces. Furthermore, we discuss the controllability of second-order neutral integrodifferential systems in finite dimensional spaces. The results are obtained by using strongly continuous cosine families of operators and the Banach fixed point theorem.

Controllability of nonlinear stochastic neutral impulsive systems

The impulsive differential equations provide a natural description of observed evolutionary processes, which are subject to short term perturbations acting instantaneously in the form of impulses. Uncertainty can be incorporated either as an expression of our lack of precise knowledge or as a true driving force. In the latter case it is useful to model the system by a stochastic or noise driven model which leads to the study of stochastic impulsive differential systems. In this paper, the notion of complete controllability for nonlinear stochastic neutral impulsive systems in finite dimensional spaces is introduced. Sufficient conditions ensuring the complete controllability of the nonlinear stochastic impulsive system are established. The results are obtained by using the Banach fixed point theorem. Several forms of integrodifferential impulsive systems are indicated. Two examples are discussed to illustrate the efficiency of the obtained results.

Analysis of quantum semigroups with GKS–Lindblad generators: II. General

Semigroups describing the time evolution of open quantum systems in finite-dimensional spaces have generators of a special form, known as Lindblad generators. These generators and the corresponding processes of time evolution are analyzed and characterized as decay, dissipation and dephasing. In relation to these processes the Hilbert space of the system is equipped with a special structure, a decomposition into a sum of mutually orthogonal subspaces. The complete set of all the stationary states and the asymptotic behavior of the evolutions are presented in detail. Some unusual special facts about invariant operators and symmetries are studied and examples are demonstrated. Perturbation theory for the structure and for the stationary states is discussed and performed in case studies.

Analysis of quantum semigroups with GKS–Lindblad generators: I. Simple generators

Semigroups describing the time evolution of open quantum systems in finite-dimensional spaces have generators of a special form, known as Lindblad generators. The simple generators, characterized by only one operator, are analyzed. The complete set of all the stationary states is presented in detail, including a formula to calculate a stationary state from the generating operator. Also the opposite task can be fulfilled, to construct an evolution leading to a prescribed stationary state.