Impulsive Differential Systems Research Articles

An analysis on approximate controllability results for impulsive fractional differential equations of order 1 < r < 2 with infinite delay using sequence method

In this work, we discuss the existence and approximate controllability results for impulsive fractional differential systems of order with infinite delay. Sectorial operator of type , the nonlinear alternative of the Leray–Schauder fixed point theorem, sequence method, and impulsive systems have been used to establish these results. First, we investigate the existence of mild solutions for impulsive fractional differential equations of order . We also establish the approximate controllability results for the nonlocal fractional delay differential equations. An example is also given to illustrate our main results.

Some stability results on non-linear singular differential systems with random impulsive moments

This paper studies the exponential stability for random impulsive non-linear singular differential systems. We established some new sufficient conditions for the proposed singular differential system by using the Lyapunov function method with random impulsive time points. Further, to validate the theoretical results' effectiveness, we finally gave two numerical examples that study with graphical illustration and an additional example involving matrices with complex entries, proving the results to be true in that case as well.

Optimal controls of impulsive fractional stochastic differential systems driven by Rosenblatt process with state‐dependent delay

AbstractIn this paper, we study a new class of noninstantaneous impulsive fractional stochastic differential systems driven by the Rosenblatt process with state‐dependent delay. We utilized the ‐resolvent family, fixed point technique, and solution operator to present the solvability of the proposed system. Further, we derived the existence of optimal multicontrol pairs for the considered system. Finally, the main results are validated with the aid of an example.

Practical exponential stability of hybrid impulsive stochastic functional differential systems with delayed impulses

SummaryThis paper is concerned with the problem of practical exponential stability for hybrid impulsive stochastic functional differential systems with delayed impulses, which comprise three classes of systems: the systems with unstable continuous stochastic dynamics and stable discrete dynamics, the systems with stable continuous stochastic dynamics and unstable discrete dynamics, and the systems where both the continuous stochastic dynamics and the discrete dynamics are stable. By using the Lyapunov‐Razumikhin approach, several new sufficient conditions for the practical exponential stability are established for each class of systems. It shows that the stabilizing and destabilizing delayed impulses that satisfy some conditions on their frequency and amplitude can stabilize the systems with unstable continuous stochastic dynamics in the practical exponential stability sense and ensure the practical exponential stability of the systems with stable continuous stochastic dynamics, respectively. Conversely, if the continuous stochastic dynamics are practically exponentially stable and the delayed impulses are stabilizing, then the systems can be practically exponentially stable regardless of the restrictions on the delayed impulses frequency and amplitude. Finally, two numerical examples are presented to illustrate the efficiency of the results.

Practical stability analysis of stochastic functional differential systems with G-Brownian motion and impulsive effects

This article is concerned with the practical stability performance of nonlinear impulsive stochastic functional differential systems driven by G-Brownian motion (G-ISFDSs). Comparing with traditional Lyapunov stability theory, practical stability can portray qualitative behavior and quantitative properties of suggested systems. By employing G-Itô formula, Lyapunov-Razumikhin approach and stochastic analysis theory, some novel conditions for pth moment practical exponential stability and quasi sure global practical uniform exponential stability of G-ISFDSs are established. The obtained results show that impulses may influence dynamic behavior of the addressed system. Two numerical examples are given to verify the validity of our developed results.

On ν-Level Interval of Fuzzy Set for Fractional Order Neutral Impulsive Stochastic Differential System

The main concern of this paper is to investigate the existence and uniqueness of a fuzzy neutral impulsive stochastic differential system with Caputo fractional order driven by fuzzy Brownian motion using fuzzy numbers with bounded ν-level intervals that are convex, normal and upper-semicontinuous. Fuzzy Itô process, Grönwall’s inequality and the Banach fixed-point theorem are employed to probe the local and global existence. An analytical example is provided to examine the theoretical results.

Existence and approximate controllability results for second-order impulsive stochastic neutral differential systems

This paper focuses on the existence and approximate controllability of second-order non-autonomous impulsive stochastic neutral differential systems in Hilbert spaces. First, by using Schauder's fixed-point theorem, stochastic analysis theory, and evolution operators, a new set of sufficient conditions are formulated, and we prove the existence of mild solutions of second-order non-autonomous impulsive stochastic neutral differential systems. Further, the result is extended to study the approximate controllability results for second-order non-autonomous impulsive stochastic neutral differential systems. Then, a set of sufficient conditions are derived for the approximate controllability of second-order non-autonomous impulsive stochastic neutral differential system by assuming the associated linear system is approximately controllable. The results are obtained with the help of Krasnoselskii's fixed point theorem. Finally, an example is given to illustrate our obtained results.

Iterative learning tracking control for second‐order nonlinear hyperbolic impulsive partial differential systems

AbstractThis paper studies the iterative learning control (ILC) for a class of second‐order nonlinear hyperbolic impulsive partial differential systems. Firstly, to follow the discontinuous desired output, a P‐type learning law is adopted, and sufficient conditions for the convergence of the tracking error is established under identified initial state value. The rigorous analysis is also given using the impulsive Gronwall inequality. Secondly, the tracking error of output trajectory is considered in systems with state initial values shifting based on an initial learning algorithm. These results of this paper show that the tracking error on the finite time interval can uniform converge to 0 as the iteration index goes to infinity if impulse number of the systems is only a finite numbers. Finally, two numerical simulation examples are given to verify the effectiveness of the theoretical results.

Approximate Controllability of Delayed Fractional Stochastic Differential Systems with Mixed Noise and Impulsive Effects

We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter H^∈(1/2,1). Using fixed point techniques, a q-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system. Moreover, under appropriate conditions, we investigate the approximate controllability of the considered system. Finally, the main results are demonstrated with an illustrative example.

Total approximate controllability of non-instantaneous impulsive fractional differential systems involving Riemann-Liouville derivatives of order <inline-formula><tex-math id="M1">$ \beta \in (1,2) $</tex-math></inline-formula>

This artifact introduces the concept of total approximate controllability in order to seek approximate controllability of the impulsive systems at break points in addition to the final point. The chosen system of inspection is a nonlinear fractional differential system affected by non-instantaneous impulses. The system is governed by Riemann-Liouville derivatives of higher-order with fixed lower limits. The appropriate integral-type initial conditions are determined differently depending on the impulsive functions. Firstly, mild solution of the concerned system is constructed, followed by a sufficient set of assumptions required to manifest the existence, uniqueness, and controllability results. Next, the definition of total approximate controllability is interposed. Further, the total approximate controllability of the concerned fractional system is established using Riemann-Liouville fractional resolvent, appropriately defined interval-wise Nemytskii operators, and an iterative technique. Lastly, an illustration is put forward to validate the proposed methodology.

Uniform Stability of Linear Delay Impulsive Differential Systems With Impulse Time Windows and Logic Choice

Uniform Stability of Linear Delay Impulsive Differential Systems With Impulse Time Windows and Logic Choice

A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order $ {r} \in (1, 2) $ with impulses

<abstract><p>In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order $ 1 < r < 2 $. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems.</p></abstract>

Multiple subharmonic solutions with prescribed minimal periods for a class of second order impulsive differential systems

In this paper, we obtain three subharmonic solutions with different prescribed minimal periods for a class of second order impulsive differential systems. Our proof is based on level estimates and the least action principle in critical point theory.

Existence of solution for an impulsive differential system with improved boundary value conditions

<abstract><p>In this paper, we study the existence of solution for a class of impulsive integro-differential equations. Different from traditional periodic and anti-periodic boundary value problems, a more general boundary condition introduced in this new system. First, we obtain some new comparison principles. Then, we obtain the expression of the solution for a class of linearized systems. Finally, the existence of extremal solutions for the new boundary value system are obtained by using the monotone iterative technique. The theoretical results obtained have wider applications in practical fields.</p></abstract>

Direct methods for singular integral equations and non-homogeneous parabolic PDEs

In this article, the author presented some applications of the Laplace, \(L^2\), and Post-Widder transforms for solving fractional Singular Integral Equations, impulsive differential equation and systems of differential equations. Finally, analytic solution for a non-homogeneous partial differential equation with non-constant coefficients is given. The obtained results reveal that the integral transform method is an effective tool and convenient.

Existence and uniqueness of periodic solution to second‐order impulsive differential equations

In this paper, existence and uniqueness of periodic solutions of second‐order semilinear impulsive differential equations are obtained. Some sufficient conditions that guarantee the existence of periodic solutions to second‐order linear nonhomogeneous impulsive differential equations are obtained. In addition, by using the appropriate fixed theorem and the theory of coincidence degree, some conditions ensuring the existence and uniqueness of periodic solution to second‐order semilinear impulsive differential systems are derived. Finally, some examples are provided to illustrate the availability of our results.

On the Analysis of a Neutral Fractional Differential System with Impulses and Delays

In this paper, we investigate the exact and approximate controllability, finite time stability, and β–Hyers–Ulam–Rassias stability of a fractional order neutral impulsive differential system. The controllability criteria is incorporated with the help of a fixed point approach. The famous generalized Grönwall inequality is used to study the finite time stability and β–Hyers–Ulam–Rassias stability. Finally, the main results are verified with the help of an example.

Iterative learning control for conformable stochastic impulsive differential systems with randomly varying trial lengths

Abstract In this paper, we introduce a new kind of conformable stochastic impulsive differential systems (CSIDS) involving discrete distribution of Bernoulli. For random discontinuous trajectories, we modify the tracking error of piecewise continuous variables by a zero-order holder. First, the improved P-type and PD α -type learning laws of the random iterative learning control (ILC) scheme are designed through global and local averaging operators. Next, we establish sufficient conditions for convergence of the tracking error in the expectation sense and prove the main results by using the impulsive Gronwall inequality and mathematical analysis tools. Finally, the theoretical results are verified by two numerical examples, and the tracking performance is compared for different conformable order of α.

An application of fixed point technique of Rothe’s-type to interpret the controllability criteria of neutral nonlinear fractional ordered impulsive system

The novel findings of this research utilizes Rothe’s fixed point technique to examine the relative controllability of neutral impulsive nonlinear differential system of arbitrary order β∈(0,1). We use pseudo-transition matrix and the technique of Laplace transform to obtain the existence results. On considering the linear form of the system to be relatively controllable as our main hypothesis, we utilize fixed point argument of Rothe’s type to establish the required conditions which are sufficiently enough for the nonlinear system to admit relative controllability. Examples are provided to validate the accuracy of the derived results and finally we apply the obtained techniques to practically model a three solution mixer.

Second-order impulsive differential systems of mixed type: oscillation theorems

In this paper, we obtain necessary and sufficient conditions for the oscillation of solutions to a second-order neutral differential equation with impulses. Two examples are provided to show the effectiveness and feasibility of the main results. Our main tool is Lebesgue’s dominated convergence theorem.