Instantaneous Impulses Research Articles

Variational method to differential equations with instantaneous and non-instantaneous impulses

The aim of this paper is to study the existence of solutions for second-order differential equations with instantaneous and non-instantaneous impulses. Applying variational method, the existence result is obtained.

The optimal behavior of solutions to fractional impulsive stochastic integro-differential equations and its control problems

In this paper, we study a new class of fractional nonlinear impulsive stochastic integro-differential equations with infinite delay in separable Hilbert spaces. Firstly, by using the measure of noncompactness, stochastic analysis theory, solution operators and suitable fixed point theorems, we prove the existence of mild solutions for these systems. Secondly, the existence of optimal mild solutions is obtained. Thirdly, we establish the controllability of the controlled fractional impulsive stochastic functional integro-differential systems with not instantaneous impulses. Finally, an example is provided to show the application of our results.

Optimality of fractional impulsive partial stochastic differential systems with analytic sectorial operators and controls

In this paper, we study a class of fractional impulsive partial stochastic differential systems with analytic sectorial operators and not instantaneous impulses in separable Hilbert spaces. Firstly, the existence of mild solutions and optimal mild solutions investigated by utilizing the theory of analytic sectorial operators, stochastic analysis, suitable fixed point theorems with the Hausdorff measure of noncompactness. Secondly, the question of exact controllability of these control systems is considered too. Finally, an example to illustrate the applications of main results is also given.

Existence of Optimal Mild Solutions and Controllability of Fractional Impulsive Stochastic Partial Integro‐Differential Equations with Infinite Delay

AbstractIn this paper, we investigate a new class of fractional impulsive stochastic partial integro‐differential equations with infinite delay in Hilbert spaces. By using the stochastic analysis theory, fractional calculus, analytic α‐resolvent operator and the fixed point technique combined with fractional powers of closed operators, we firstly give the existence of of mild solutions and optimal mild solutions for the these equations. Next, the controllability of the controlled fractional impulsive stochastic partial integro‐differential systems with not instantaneous impulses is presented. Finally, examples are also given to illustrate our results.

Existence of Mild Solution for Mixed Volterra–Fredholm Integro Fractional Differential Equation with Non-instantaneous Impulses

We establish a set of sufficient conditions for the existence of mild solution of a class of fractional mixed integro differential equation with not instantaneous impulses. The results are obtained by establishing two theorems by using semigroup theory, Banach fixed point theorem and Krasnoselskii’s fixed point theorem. Two examples are presented to validate the results of the theorems.

Existence and stability results for partial implicit fractional differential equations with not instantaneous impulses

Existence and stability results for partial implicit fractional differential equations with not instantaneous impulses

A class of fractional order systems with not instantaneous impulses

A class of fractional order systems with not instantaneous impulses

Mittag–Leffler Stability for Impulsive Caputo Fractional Differential Equations

The stability properties of Caputo fractional differential equations with impulses are studied. Both types of impulses, non-instantaneous impulses as well as instantaneous impulses are considered. The two approaches in the literature for the interpretation ofsolutions of impulsive Caputo fractional differential equations are presented and discussed. A generalization of Mittag–Leffler stability with respect to both types of impulses is given.

Stability results for fractional differential equations with state-dependent delay and not instantaneous impulses

Abstract In this paper, we shall present some uniqueness and Ulam’s type stability concepts for the Darboux problem of partial functional differential equations with not instantaneous impulses and state-dependent delay in Banach spaces. Some examples are also provided to illustrate our results.

Optimal Controls of Systems Governed by Semilinear Fractional Differential Equations with Not Instantaneous Impulses

This paper is concerned on optimal control problems for systems governed semilinear fractional differential equations with not instantaneous impulses in the infinite dimensional spaces. We utilize fractional calculus, semigroup theory and fixed point approach to present the solvability of the corresponding control system by using the new introduced concept of mild solutions. Next, we give the existence result of optimal controls for Lagrange problem under the suitable conditions. Finally, an example is given to illustrate the effectiveness of our results.

Variational approach to differential equations with not instantaneous impulses

In this note we introduce the concept of a weak solution for a linear equation with not instantaneous impulses. We use the classical Lax–Milgram Theorem to reveal the variational structure of the problem and get the existence and uniqueness of weak solutions as critical points. This will allow us in the future to deal with the corresponding nonlinear problems.

Prompt GPS TEC response to magnetospheric compression

AbstractA new type of total electron content (TEC) variation was observed when an interplanetary shock impacted on the Earth's magnetosphere on 17 March 2015. With hundreds of ground‐based Global Positioning System (GPS) receivers, instantaneous TEC impulses were detected in the signals from four GPS satellites cruising in the dayside equatorial magnetosphere. Despite the small amplitude (0.3 TECU), the impulses were synchronously registered by receivers spreading from low to middle latitudes on the ground; they lasted for several minutes, during which period the interplanetary magnetic field (IMF) kept northward. The TEC impulses therefore discriminate themselves from usual traveling ionospheric disturbances and exclude possible effects from the southward IMF‐driven magnetic storm. We suggest that the TEC variation is caused by shock‐induced magnetospheric compression, which moves plasma earthward in the dayside plasmasphere. As a result, some plasma outside of GPS orbit (4.2 RE) is moved to the inside and contributes to the plasma content traversed by GPS raypath. The GPS TEC technique thus exhibits an unprecedented capability to capture small tremor of the magnetosphere, and with the dense receiver network on the ground it can be a feasible tool for remote sensing of the plasma dynamics around 4.2 RE.

On General Solution for Fractional Differential Equations with Not Instantaneous Impulses

On General Solution for Fractional Differential Equations with Not Instantaneous Impulses

Some stability concepts for abstract fractional differential equations with not instantaneous impulses

Some stability concepts for abstract fractional differential equations with not instantaneous impulses

Critical Response of Free-Standing Rocking Blocks to the Intense Phase of an Earthquake

The paper deals with a simplified approach to assess the critical response of slender rigid blocks to the intense phase of an earthquake. First, an artificial limit accelerogram is proposed to represent the most unfavorable effects of this phase. It consists of a sequence of instantaneous impulses, all applied right after the impact of the blocks on the ground, that continuously add energy to the system and cause rocking resonance. The results highlight to what extent the ground motion details and the system parameters can influence the rocking response. A secondary sequence of intermediate impulses is then introduced to reduce the resonance effects and to cover a broad range of conditions. It is shown that the intermediate impulses have a stabilizing effect, increasing with their amplitude. Numerical analyses are performed to highlight the most significant aspects of the proposed approach.

The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

Dynamic analysis of gas permeable blasting mat as geometrically nonlinear system with unilateral constraints

Blasting mat is used to prevent flyrock accidents, decrease environmental impacts and expand the range of projects where blasting operations could be carried out. Transformable gas permeable blasting mat is made from worn out tires of heavy trucks bound together with chains, ropes or cables.Dynamic model of blasting mat is proposed in this research. It is represented by concentrated masses interacting with unilateral foundation and by elastic weightless incompressible elements. The problem solving involves combination of two types of nonlinearity: geometrical nonlinearity associated with large displacements and contact nonlinearity which emanates from the fact that contact region changes due to deformation of the elements. Explosive loading is simulated by instantaneous impulses applied at different times to different masses. The system behavior under various sequences of explosions is of special interest.Numerical experiments showed that axial forces in elastic elements and maximum flight height of the masses depend on detonation sequence of the blast holes. Employed mathematical model was verified by full-scale experiment.

Existence and Controllability Results for a New Class of Impulsive Stochastic Partial Integro‐Differential Inclusions with State‐Dependent Delay

AbstractIn this paper, we study a new class of impulsive stochastic partial integro‐differential inclusions with state‐dependent delay in separable Hilbert spaces. Firstly, by using stochastic analysis theory, analytic resolvent operators, fractional powers of closed operators and suitable fixed point theorems, we prove the existence of mild and extremal mild solutions for these systems in the α‐norm. Secondly, we establish the controllability of the controlled stochastic partial integro‐differential inclusions with not instantaneous impulses. The results are obtained under the mixed Lipschitz and Carathéodory conditions. Finally, an example is provided to show the application of our results.

On the fractional differential equations with not instantaneous impulses

AbstractBased on some previous works, an equivalent equations is obtained for the differential equations of fractional-orderq∈(1, 2) with non-instantaneous impulses, which shows that there exists the general solution for this impulsive fractional-order systems. Next, an example is used to illustrate the conclusion.

Upper and lower solutions method for partial discontinuous fractional differential inclusions with not instantaneous impulses

Upper and lower solutions method for partial discontinuous fractional differential inclusions with not instantaneous impulses