Fractional-order Differential Inclusions Research Articles

Global Mittag–Leffler stabilization of fractional-order complex-valued memristive neural networks

This paper presents the theoretical results about global Mittag–Leffler stabilization for a class of fractional-order complex-valued memristive neural networks with the designed two types of control rules. As the extension of fractional-order real-valued memristive neural networks, fractional-order complex-valued memristive neural networks have complex-valued states, synaptic weights, and the activation functions. By utilizing the set-valued maps, a generalized fractional derivative inequality as well as fractional-order differential inclusions, several stabilization criteria for global Mittag–Leffler stabilization of fractional-order complex-valued memristive neural networks are established. A numerical example is provided here to illustrate our theoretical results.

Global dissipativity and quasi-synchronization of asynchronous updating fractional-order memristor-based neural networks via interval matrix method

This paper investigates the global dissipativity and quasi-synchronization of asynchronous updating fractional-order memristor-based neural networks (AUFMNNs) via interval matrix method. First, a new class of FMNNs named AUFMNNs is proposed for the first time, in which the switching jumps are asymmetric. In other words, each memristive connection weight is updated based on its own channel and hence the number of the subsystems increases significantly from 2n to 22n2. Under the framework of fractional-order differential inclusions, the proposed AUFMNNs can be regarded as a system with interval parameters. Then, the global dissipativity criterion is established by constructing appropriate Lyapunov function in combination with the estimates of 2-norm for interval matrices and some fractional-order differential inequalities. In addition, for drive-response AUFMNNs with mismatched parameters, the problem of quasi-synchronization is explored via linear state feedback control. It has been shown that complete synchronization between two AUFMNNs cannot be achieved via linear feedback control and that the synchronization error bound can be controlled within a relatively small level by selecting suitable control parameters. Finally, three numerical examples are given to demonstrate the effectiveness and the improvement of the obtained results.

Improved quasi-synchronization criteria for delayed fractional-order memristor-based neural networks via linear feedback control

Abstract This paper is concerned with the quasi-synchronization of two delayed fractional-order memristor-based neural networks (FMNNs) with mismatched switching jumps via linear feedback control. The concept of asynchronous switching time interval (ASTI) is introduced first to describe when the drive-response FMNNs update their connection weights asynchronously. Under the framework of fractional-order differential inclusions, two improved quasi-synchronization criteria, expressed by algebraic conditions and LMIs conditions respectively, are established by constructing appropriate Lyapunov functionals in combination with some fractional-order differential inequalities. Different from most previously published works, the synchronization error bound can be estimated without requiring the bound of chaotic trajectories. In addition, it has been shown that the degree of mismatch between the switching jumps has an important influence on the distribution of ASTI as well as the practical synchronization error. Finally, two numerical examples are given to verify the validity and feasibility of the obtained results.

Quasi‐synchronisation of fractional‐order memristor‐based neural networks with parameter mismatches

This study addresses the problem of quasi-synchronisation of fractional-order memristor-based neural networks (FMNNs) with time delay in the presence of parameter mismatches. Under the framework of fractional-order differential inclusions and set-valued maps, quasi-synchronisation of delayed FMNNs is discussed and quasi-synchronisation criteria are established by means of constructing suitable Lyapunov function, together with introducing some fractional-order differential inequalities. A new lemma on the estimate of Mittag–Leffler function is derived first, which extends the application of Mittag–Leffler function and plays a key role in the estimate of synchronisation error bound. Then, linear state feedback combined with delayed state feedback control law is designed, which guarantees that for a predetermined synchronisation error bound, quasi-synchronisation of two FMNNs with mismatched parameters will be achieved provided that the feedback gains satisfy the newly-proposed criteria. The obtained results extend and improve some previous published works on synchronisation of FMNNs. Finally, two numerical examples are given to demonstrate the effectiveness of the obtained results.

On semilinear fractional order differential inclusions in Banach spaces

On semilinear fractional order differential inclusions in Banach spaces

Global Mittag-Leffler Stabilization of Fractional-Order Memristive Neural Networks.

According to conventional memristive neural network theories, neurodynamic properties are powerful tools for solving many problems in the areas of brain-like associative learning, dynamic information storage or retrieval, etc. However, as have often been noted in most fractional-order systems, system analysis approaches for integral-order systems could not be directly extended and applied to deal with fractional-order systems, and consequently, it raises difficult issues in analyzing and controlling the fractional-order memristive neural networks. By using the set-valued maps and fractional-order differential inclusions, then aided by a newly proposed fractional derivative inequality, this paper investigates the global Mittag-Leffler stabilization for a class of fractional-order memristive neural networks. Two types of control rules (i.e., state feedback stabilizing control and output feedback stabilizing control) are designed for the stabilization of fractional-order memristive neural networks, while a list of stabilization criteria is established. Finally, two numerical examples are given to show the effectiveness and characteristics of the obtained theoretical results.

On Noncompact Fractional Order Differential Inclusions with Generalized Boundary Condition and Impulses in a Banach Space

We provide existence results for a fractional differential inclusion with nonlocal conditions and impulses in a reflexive Banach space. We apply a technique based on weak topology to avoid any kind of compactness assumption on the nonlinear term. As an example we consider a problem in population dynamic described by an integro-partial-differential inclusion.

Existence and Ulam Stability for Partial Impulsive Discontinuous Fractional Differential Inclusions in Banach Algebras

In this paper, we investigate some existence and Ulam’s type stability concepts of fixed point inclusions for a class of partial discontinuous fractional-order differential inclusions with impulses in Banach Algebras. Our results are obtained using weakly Picard operators theory.

Ulam Stability for Partial Fractional Differential Inclusions via Picard Operators Theory

In the present paper, we investigate, using the Picard operator technique, some existence and Ulam type stability results for the Darboux problem associated to some partial fractional order differential inclusions.

Some Remarks on a Multi Point Boundary Value Problem for a Fractional Order Differential Inclusion

We study a multi point boundary value problem for a fractional order differential inclusion involving a nonconvex set-valued map. We establish a Filippov type existence theorem and we prove the arcwise connectedness of the solution set of the problem considered.

On a class of fractional order differential inclusions with infinite delays

Our aim is to study fractional order differential inclusions with infinite delays in Banach spaces. We impose the regularity condition on multivalued nonlinearity in terms of measures of noncompactness to get the existence result. Some properties of the solution map are proved.

On a Multipoint Boundary Value Problem for a Fractional Order Differential Inclusion on an Infinite Interval

We investigate the existence of solutions for the following multipoint boundary value problem of a fractional order differential inclusionD0+αut+Ft,ut,u′t∋0,0<t<+∞,u0=u′0=0,Dα-1u+∞-∑i=1m-2‍βiuξi=0, whereD0+αis the standard Riemann-Liouville fractional derivative,2<α<3,0<ξ1<ξ2<⋯<ξm-2<+∞, satisfies0<∑i=1m-2‍βiξiα-1<Γ(α), and F:[0,+∞)×ℝ×ℝ→𝒫(ℝ)is a set-valued map. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or nonconvex values.

On a multi point boundary value problem for a fractional order differential inclusion

The existence of solutions for a multi point boundary value problem of a fractional order differential inclusion is investigated. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.

Stabilization of chaos systems described by nonlinear fractional-order polytopic differential inclusion

In this paper, sliding mode control is utilized for stabilization of a particular class of nonlinear polytopic differential inclusion systems with fractional-order-0 < q < 1. This class of fractional order differential inclusion systems is used to model physical chaotic fractional order Chen and Lu systems. By defining a sliding surface with fractional integral formula, exploiting the concept of the state space norm, and obtaining sufficient conditions for stability of the sliding surface, a special feedback law is presented which enables the system states to reach the sliding surface and consequently creates a sliding mode control. Finally, simulation results are used to illustrate the effectiveness of the proposed method.

Fractional Order Differential Inclusions via the Topological Transversality Method

El objetivo de este trabajo es presentar nuevos resultados sobre la existencia de soluciones para una clase de problemas de contorno para inclusiones diferenciales derivados de la participacion de Caputo fraccionada. Nuestro enfoque se basa en el metodo de la transversalidad topologica.

A Filippov’s Theorem, Some Existence Results and the Compactness of Solution Sets of Impulsive Fractional Order Differential Inclusions

In this paper, we first present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form, $$\begin{array}{lll} \quad \qquad D^{\alpha}_{*}y(t) & \in & F(t, y(t)), \quad\; {\rm a.e.}\ t\, \in \, J{\backslash} \{t_{1}, \ldots, t_{m}\}, \ \alpha\, \in \, (0,1], \\ y(t^{+}_{k}) - y(t^{-}_{k}) & = & I_{k}(y(t^{-}_{k})), \quad k = 1, \ldots, m, \\ \qquad \qquad y(0) & = & a,\end{array}$$ where J = [0, b], \({D^{\alpha}_{*}}\) denotes the Caputo fractional derivative and F is a set-valued map. The functions Ikcharacterize the jump of the solutions at impulse points tk (\({k = 1, \ldots , m}\)). In addition, several existence results are established, under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on a nonlinear alternative of Leray-Schauder type and on Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of solution sets is also investigated.

Stabilization of multi-input hybrid fractional-order systems with state delay

In this paper, the stabilization of a particular class of multi-input linear systems of fractional order differential inclusions with state delay using variable structure control is considered. First, the sliding surface with a fractional order integral formula is defined, and then the sufficient conditions for stability of the sliding surface are derived. Also, the concepts related to sliding control stabilization of differential inclusion systems with integer order are extended to differential inclusion systems with fractional order 0 < q < 1 . Furthermore, for stabilization of fractional order systems with delay in the state space, a feedback and the concept of the norm of a state space is exploited. Finally, simulation results are provided to show the effectiveness of the proposed method.

On a nonlinear fractional order differential inclusion

The existence of solutions for a nonlinear fractional order differential inclusion is investigated. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.

The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses

The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses