On the approximate controllability of semilinear fractional differential systems

Fractional differential equations have wide applications in science and engineering. In this paper, we consider a class of control systems governed by the semilinear fractional differential equations in Hilbert spaces. By using the semigroup theory, the fractional power theory and fixed point strategy, a new set of sufficient conditions are formulated which guarantees the approximate controllability of semilinear fractional differential systems. The results are established under the assumption that the associated linear system is approximately controllable. Further, we extend the result to study the approximate controllability of fractional systems with nonlocal conditions. An example is provided to illustrate the application of the obtained theory.     

Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces

We study the existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a Hilbert space. The results are obtained by using semigroup theory, fractional calculus and Schauder’s fixed point theorem. Multi-delay controls and a fractional nonlocal condition are introduced. Furthermore, we present an appropriate set of sufficient conditions for the considered fractional nonlocal multi-delay control system to be approximately controllable. An example to illustrate the abstract results is given.     

The cost of approximate controllability for semilinear heat equations

We consider the semilinear heat equation with globally Lipschitz non-linearity involving gradient terms in a bounded domain of R^n. In this paper, we obtain explicit bounds of the cost of approximate controllability, i.e., of the minimal norm of a control needed to control the system approximately. The methods we used combine global Carleman estimates, the variational approach to approximate controllability and Schauder＇s fixed point theorem.     

Approximate controllability for abstract measure differential systems

In this paper, we investigate approximate controllability for abstract measure differential systems based on generalizing knowledge for ordinary differential systems. We first introduce new concepts of the reachable set and approximate controllability for abstract measure differential systems. Then based on the nonlinear alternative for α-condensing mapping in Banach space, we present sufficient conditions of approximate controllability for a class of abstract measure differential systems. Our results in dealing with approximate controllability are less conservative than those in the previous literature. Finally, an example is given to illustrate the availability of our results for approximate controllability.     

Approximate controllability results of semilinear integrodifferential equations with infinite delays

We study the approximate controllability of control systems governed by a class of semilinear integrodifferential equations with infinite delays. Sufficient conditions for approximate controllability of semilinear control systems are established provided the approximate controllability of the corresponding linear control systems. The results are obtained by using fixed-point theorems and semigroup theory. As an illustration of the application of the approximate controllability results, a simple example is provided.     

Existence of solutions for a singular system of nonlinear fractional differential equations

In this paper, we establish the existence of positive solutions for a singular system of nonlinear fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Green’s function and its corresponding properties, we transform the derivative systems into equivalent integral systems. The existence is based on a nonlinear alternative of Leray-Schauder type and Krasnoselskii’s fixed point theorem in a cone.     

Approximate solution of fractional integro-differential equations by Taylor expansion method

In this paper, Taylor expansion approach is presented for solving (approximately) a class of linear fractional integro-differential equations including those of Fredholm and of Volterra types. By means of the mth-order Taylor expansion of the unknown function at an arbitrary point, the linear fractional integro-differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions. This method gives a simple and closed form solution for a linear fractional integro-differential equation. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.     

Relative controllability of nonlinear switched fractional systems

This article investigates relative controllability of nonlinear switched fractional systems (SFSs). With the aid of Mittag-Leffler functions and invariant subspace theory, two sufficient and necessary conditions for corresponding linear SFSs are first established. Then, piecewise continuous control functions and a novel nonlinear operator are constructed to overcome the difficulties arising from switching rules, nonlinearity, and fractional derivatives. Under different constraints on nonlinear functions, two controllability conditions depending on system parameters for nonlinear SFSs are proposed by applying Schauder's and Banach's fixed point theorems, respectively. The obtained criteria may well show the influence of coefficient matrices, fractional derivatives, and switching rules on relative controllability. In addition, our proposed method is also applicable for integer-order switched systems. Finally, two simulated examples are worked out to verify the theoretical results.     

Approximate controllability of impulsive differential equations with state-dependent delay

In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, impulsive differential equations have been used to describe the system model. In this article, the problem of approximate controllability for nonlinear impulsive differential equations with state-dependent delay is investigated. We study the approximate controllability for nonlinear impulsive differential system under the assumption that the corresponding linear control system is approximately controllable. Using methods of functional analysis and semigroup theory, sufficient conditions are formulated and proved. Finally, an example is provided to illustrate the proposed theory.     

Null controllability for a fourth order parabolic equation

In the paper, the null interior controllability for a fourth order parabolic equation is obtained. The method is based on Lebeau-Rabbiano inequality which is a quantitative unique continuation property for the sum of eigenfunctions of the Laplacian.     

Anti-periodic fractional boundary value problems

We study a class of anti-periodic boundary value problems of fractional differential equations. Some existence and uniqueness results are obtained by applying some standard fixed point principles. Several examples are given to illustrate the results.     

Approximate controllability results for impulsive neutral differential inclusions of Sobolev-type with infinite delay

The paper is mainly concerned with the approximate controllability results for a class of impulsive neutral differential control systems. First, we establish a set of sufficient conditions for the approximate controllability for a class of impulsive differential inclusions of Sobolev-type with infinite delay. With the help of semigroup theory and Bohnenblust-Karlin's fixed point theorem, we prove our main results. Then, we extend the result to study the approximate controllability for a class of impulsive neutral differential inclusions of Sobolev-type with infinite delay. Finally, an example is also presented to illustrate our theoretical results.     

Complete controllability for abstract measure differential systems

In this paper, we investigate the complete controllability for abstract measure differential systems. Firstly, we introduce several new concepts about complete controllability for abstract measure differential systems. Then, on the basis of the Sadovskii fixed‐point theorem, we give sufficient conditions for complete controllability for a class of abstract measure differential systems. The compactness of the semigroup generated by some operator is unnecessary in this paper, and we show that our results, dealing with complete controllability problem for an ordinary differential system in infinite‐dimensional Banach space, are also less conservative than that in the previous literature. Copyright © 2012 John Wiley & Sons, Ltd.     

Uniform ensemble controllability for one-parameter families of time-invariant linear systems

In this paper we derive necessary as well as sufficient conditions for approximate controllability of parameter-dependent linear systems in the supremum norm. Using tools from complex approximation theory, we prove the existence of parameter-independent open-loop controls that steer the zero initial state of an ensemble of linear systems uniformly to a prescribed family of terminal states. New necessary conditions for uniform ensemble controllability of single-input systems are derived. Our results extend earlier ones of Li for ensemble controllability of linear systems.     

Approximate controllability results for analytic resolvent integro-differential inclusions in Hilbert spaces

In this work, we consider a nonlinear resolvent integro-differential evolution inclusions in Hilbert spaces. This paper deals with the approximate controllability for nonlinear resolvent integro-differential inclusions in Hilbert spaces. We use Bohnenblust–Karlin's fixed-point theorem to establish a set of sufficient conditions for the approximate controllability for nonlinear resolvent integro-differential inclusions in Hilbert spaces. Further, we extend the result to study the approximate controllability concept with non-local conditions. An example is presented to demonstrate the obtained theory.     

Attractivity of fractional functional differential equations

In this paper, some attractivity results for fractional functional differential equations are obtained by using the fixed point theorem. By constructing equivalent fractional integral equations, research on the attractivity of fractional functional and neutral differential equations is skillfully converted into a discussion about the existence of fixed points for equivalent fractional integral equations. Two examples are also provided to illustrate our main results.     

分数阶扩散系统子区域能控性研究的现状与发展

近年来,随着分数阶微积分理论研究的不断深入,分数阶扩散系统因更有效描述自然界和工程中的反常扩散现象而引起了学者们的广泛关注,并取得了较为丰硕的研究成果.然而,据作者所知,目前鲜有文献总结分数阶扩散系统子区域能控性研究的相关内容.因此,本文首先详细综述分数阶扩散系统能控性、子区域能控性的研究进展,并予以分析;然后给出仍需解决的问题和可能的研究机遇.