Impulsive Differential Inclusions Research Articles

Existence and Stability of Solutions for p-Proportional ω-Weighted κ-Hilfer Fractional Differential Inclusions in the Presence of Non-Instantaneous Impulses in Banach Spaces

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the p-proportional ω-weighted κ-Hilfer derivative includes an exponential function, Da,λσ,ρ,p,κ,ω, and then we consider a non-instantaneous impulse differential inclusion containing Da,λσ,ρ,p,κ,ω with order σ∈(1,2) and of kind ρ∈[0,1] in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the (p,ω,κ)-generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the (p,ω,κ)-generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator Da,λσ,ρ,p,κ,ω, our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings.

Approximate controllability results for second-order non-local neutral impulsive differential inclusions with damping

This paper is mainly concerned with the approximate controllability of damped second-order neutral impulsive differential inclusions with non-local conditions in Banach spaces. We study our primary outcomes by using the theoretical concepts about cosine and sine functions of operators, Bohnenblust and Karlin's fixed point theorem. Using principles and ideas from the theory of the cosine family of operators and the fixed-point approach, we verify the existence of mild solutions for the given system. A new set of sufficient conditions is formulated and proved for the approximate controllability of second-order differential inclusions under the assumption that the associated linear part of the system is approximately controllable. Finally, an application is presented to illustrate our theoretical results.

Antiperiodic Solutions for Impulsive ω-Weighted ϱ–Hilfer Fractional Differential Inclusions in Banach Spaces

In this article, we construct sufficient conditions that secure the non-emptiness and compactness of the set of antiperiodic solutions of an impulsive fractional differential inclusion involving an ω-weighted ϱ–Hilfer fractional derivative, D0,tσ,v,ϱ,ω, of order σ∈(1,2), in infinite-dimensional Banach spaces. First, we deduce the formula of antiperiodic solutions for the observed problem. Then, we give two theorems regarding the existence of these solutions. In the first, by using a fixed-point theorem for condensing multivalued functions, we show the non-emptiness and compactness of the set of antiperiodic solutions; and in the second, by applying a fixed-point theorem for contraction multivalued functions, we prove the non-emptiness of this set. Because many types of famous fractional differential operators are particular cases from the operator D0,tσ,v,ϱ,ω, our results generalize several recent results. Moreover, there are no previous studies on antiperiodic solutions for this type of fractional differential inclusion, so this work is novel and interesting. We provide two examples to illustrate and support our conclusions.

Viability for impulsive stochastic differential inclusions driven by fractional Brownian motion

In this paper, we prove necessary and sufficient conditions for the viability for an impulsive stochastic functional differential inclusion driven by a fractional Brownian motion. The viable property is of interest since it reflects the stability of the model under consideration. The fractional Brownian motion provides a memory effect to the model. Whereas, the appearance of the impulsive factor introduces jumps to the solutions and is new to the analysis of this type. Hence our results are new even in the special case of stochastic differential equation setting.

Iterative learning based convergence analysis for nonlinear impulsive differential inclusion systems with randomly varying trial lengths

SummaryThis paper studies the finite‐time tracking problem for nonlinear impulsive differential inclusion systems with randomly varying trial lengths. First, we convert the set‐valued mapping in the differential inclusion systems to single‐valued mapping by a Steiner‐type selector. For the tracking problem of random discontinuous output trajectories, this paper defines a piecewise continuous variable by zero‐order holder to correct the tracking error of segmented continuity. Then, we introduce the average operator with forgetting factor to design three novel learning schemes, and establish convergence results by using the mathematical analysis tools such as impulsive Gronwall inequality and ‐norm. Finally, a numerical example verifies the validity of the theoretical results, and we compare the tracking performance of the output trajectories for different forgetting factors.

The Existence of Solutions for w-Weighted ψ-Hilfer Fractional Differential Inclusions of Order μ ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces

In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted ψ-Hilfer fractional derivative, D0,tσ,v,ψ,w,of order μ∈(1,2), in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving D0,tσ,v,ψ,w of order μ ∈(1,2) in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the w-weighted ψ-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator D0,tσ,v,ψ,w includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results.

Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $ in infinite dimensional Banach spaces

<abstract><p>In this paper, we improved recent results on the existence of solutions for nonlinear fractional boundary value problems containing the Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $. We also derived the exact relations between these fractional boundary value problems and the corresponding fractional integral equations in infinite dimensional Banach spaces. We showed that the continuity assumption on the nonlinear term of these equations is insufficient, give the derived expression for the solution, and present two results about the existence and uniqueness of the solution. We examined the case of impulsive impact and provide some sufficiency conditions for the existence and uniqueness of the solution in these cases. We also demonstrated the existence and uniqueness of anti-periodic solution for the studied problems and considered the problem when the right-hand side was a multivalued function. Examples were given to illustrate the obtained results.</p></abstract>

On the solution of T−controllable abstract fractional differential equations with impulsive effects

In this research article, we delimitate the definition of mild solution for abstract fractional differential equations with state-dependent delay (AFDEw/SDD) of order α ∈(1,2) with impulsive effects and compare the solution to the second-order impulsive differential equations. Further, we obtain sufficient conditions of the existence of mild solution for instantaneous and non-instantaneous impulsive fractional functional differential inclusions with state-dependent delay (IFDIw/SDD) using the multi-valued fixed point theory and operator techniques. Furthermore, we study the trajectory controllability (T−controllability) of the AFDEw/SDD. At last, we present some examples to illustrate the sufficient conditions involving partial and ordinary derivatives.

Quantized iterative learning control for impulsive differential inclusion systems with data dropouts

This paper studies the quantized iterative learning control with encoding–decoding mechanism of a class of impulsive differential inclusion systems with random data dropouts. First, the set-valued mappings in the differential inclusion systems are transformed into single-valued mappings by using the Steiner-type selector. Then, a learning algorithm based on the intermittent update principle is designed to address the data asynchronism problem caused by two-sided data dropouts. If the data are successfully transmitted at the actuator and measurement sides, then the control input is effectively updated. Furthermore, a suitable scaling sequence is introduced to ensure the system output to achieve zero-error tracking performance for a desired trajectory. An upper bound of the quantization level is determined such that the quantization error is always bounded. The results show that the quantization method reduces the burden of network communication at the cost of increasing the amount of computation, and the learning algorithm does not require the data dropouts to satisfy a certain probability distribution. Finally, the effectiveness of the learning algorithm is verified by numerical simulations of the switched reluctance motor system.

Approximate Optimal Control of Fractional Impulsive Partial Stochastic Differential Inclusions Driven by Rosenblatt Process

Approximate Optimal Control of Fractional Impulsive Partial Stochastic Differential Inclusions Driven by Rosenblatt Process

Trajectory controllability of Clarke subdifferential type Hilfer fractional stochastic differential inclusion with non-instantaneous impulsive effects and deviated argument

This manuscript is devoted to analyse the solvability and trajectory controllability of Hilfer fractional non-instantaneous impulsive stochastic differential inclusion with Clarke subdifferential and deviated argument. The proposed Hilfer fractional impulsive inclusion system’s solvability in Hilbert space is established by employing fractional calculus, multivalued analysis, stochastic analysis, semigroup theory and a multivalued fixed point theorem. Furthermore, under some suitable assumptions, the considered system’s trajectory controllability is established using a generalized Gronwall’s inequality. At last, an example is provided to verify the developed theoretical results.

Viability and Filippov-type Lemma for Stieltjes Differential Inclusions

We prove a viability result for differential inclusions involving the Stieltjes derivative with respect to a left-continuous non-decreasing function with time dependent state constraints. A tangential condition using a generalized notion of the contingent derivative is imposed. Classical viability results (for usual differential inclusions) are thus generalized and, at the same time, the gate to new viability results for difference inclusions, impulsive differential inclusions or dynamic inclusions on time scales is open. As a consequence, in the particular case where the state constraint is a tube, a Filippov-type lemma is obtained for this very general setting, of differential problems driven by Stieltjes derivatives.

Mild solutions for impulsive fractional differential inclusions with Hilfer derivative in Banach spaces

Mild solutions for impulsive fractional differential inclusions with Hilfer derivative in Banach spaces

Topological Properties of Solution Sets for τ-Fractional Non-Instantaneous Impulsive Semi-Linear Differential Inclusions with Infinite Delay

The knowledge of fractional calculus can be useful in developing models that allow us to better understand and manage some phenomena. In the present paper, we study the topological structure of the mild solution set for a semi-linear differential inclusion containing the τ-Caputo fractional derivative in the presence of non-instantaneous impulses and an infinite delay. We demonstrate that this set is non-empty and an Rδ-set. We use a recent result regarding the existence of solutions for τ-Caputo fractional semi-linear differential inclusions. Unlike many results, we do not suppose that the generating semigroup is compact. An illustrative example is given as an application of our results.

Parametric Topological Entropy for Multivalued Maps and Differential Inclusions with Nonautonomous Impulses

The main purpose of this paper is to investigate a parametric topological entropy for impulsive differential inclusions on tori. In this way, besides other matters, we would like to extend our recent results concerning impulsive differential equations as well as those on “nonparametric” topological entropy to impulsive differential inclusions. Parametric topological entropy, which is usually called a topological entropy for nonautonomous dynamical systems, is considered here via the compositions of associated multivalued Poincaré translation operators with the single-valued time-dependent impulsive maps. On compact polyhedra and, in particular on tori, parametric topological entropy for families of admissible multivalued maps can be estimated from below by means Ivanov-type inequality in terms of the asymptotic Nielsen and Lefschetz numbers which are, unlike the topological entropy, homotopy invariants. In the scalar case, an effective criterion for a positive parametric topological entropy can be given by topological degree arguments for equi-continuous impulsive maps. In a single-valued nonparametric case, a positive topological entropy usually signifies topological chaos. Some simple illustrative examples are provided.

Relaxation result for differential inclusions with Stieltjes derivative

The aim of this paper is to provide a Filippov-Ważewski Relaxation Theorem for the very general setting of Stieltjes differential inclusions. New relaxation results can be deduced for generalized differential problems, for impulsive differential inclusions with multivalued impulsive maps and possibly countable impulsive moments and also for dynamic inclusions on time scales.

Exponential stabilization of nonlinear systems under saturated control involving impulse correction

This paper studies the problem of locally exponential stabilization (LES) of nonlinear systems under a class of hybrid control in the framework of actuator saturation, where the limitation of actuator saturation on both continuous state feedback control and impulsive control are fully considered. Based on impulsive control theory and differential inclusion approach, some sufficient conditions for LES are derived, where a novel set inclusion relation is proposed to handle the double saturation nonlinearities. Different from the existing results that each part of saturated hybrid control (SHC) is required to stabilize the system individually, our results relax the requirement by making full use of the correction effect of the impulse. Moreover, the maximum of estimation of domain of attraction is obtained by a convex optimal problem and corresponding algorithm. The result is applied to the robustness for a class of nonlinear systems. Finally, the validity of the results is shown by two examples and their simulations, where the synchronization problem of Chua’s oscillator is illustrated in the framework of actuator saturation.

Mild solutions for nonlinear impulsive functional differential inclusions with nonlocal conditions

Mild solutions for nonlinear impulsive functional differential inclusions with nonlocal conditions

Nonlocal impulsive differential equations and inclusions involving Atangana-Baleanu fractional derivative in infinite dimensional spaces

<abstract><p>The aim of this paper is to derive conditions under which the solution set of a non-local impulsive differential inclusions involving Atangana-Baleanu fractional derivative is a nonempty compact set in an infinite dimensional Banach spaces. Existence results for solutions in the presence of instantaneous or non-instantaneous impulsive effect are given. We considered the case where the right hand side is either a single valued function, or a multifunction. This generalizes recent results to the case when there are impulses, the right hand side is a multifunction, and where the dimension of the space is infinite. Examples are given to illustrate the effectiveness of the established results.</p></abstract>

Existence and controllability of fractional evolution inclusions with impulse and sectorial operator

Many evolutionary operations fromdiverse fields of engineering and physical sciences go through abrupt modifications of state at specific moments of time among periods of non-stop evolution. These operations are more conveniently modeled via impulsive differential equations and inclusions. In this work, firstly we address the existence of mild solutions for nonlocal fractional impulsive semilinear differential inclusions related to Caputo derivative in Banach spaces when the linear part is sectorial. Secondly, we determine the enough, conditions for the controllability of the studied control problem. We apply effectively fixed point theorems, contraction mapping, multivalued analysis and fractional calculus. Moreover, we enhance our results by introducing an illustrative examples.