Impulsive Differential Systems Research Articles

Optimization of exact controllability for fractional impulsive partial stochastic differential systems via analytic sectorial operators

Abstract In this paper, we consider the optimization problems of exact controllability for a new class of fractional impulsive partial stochastic differential systems with state-dependent delay in Hilbert spaces. By utilizing suitable fixed point approach without imposing severe compactness condition on the operators, the theory of analytic sectorial operators, stochastic analysis, and the Hausdorff measure of noncompactness, some sufficient conditions are derived for achieving the required results. Finally, an example is provided to illustrate the obtained theory.

Stability of nonlinear variable-time impulsive differential systems with delayed impulses

This paper investigates the stability of nonlinear variable-time impulsive differential systems with delayed impulses. By using the comparison principle, the Lyapunov method and inequality techniques, some sufficient conditions for quasi-uniformly asymptotic stability and quasi-exponential stability of a given solution of variable-time impulsive differential systems with delayed impulses are established.

Heteroclinic Chaotic Threshold in a Nonsmooth System with Jump Discontinuities

In smooth systems, the form of the heteroclinic Melnikov chaotic threshold is similar to that of the homoclinic Melnikov chaotic threshold. However, this conclusion may not be valid in nonsmooth systems with jump discontinuities. In this paper, based on a newly constructed nonsmooth pendulum, a kind of impulsive differential system is introduced, whose unperturbed part possesses a nonsmooth heteroclinic solution with multiple jump discontinuities. Using the recursive method and the perturbation principle, the effects of the nonsmooth factors on the behaviors of the nonsmooth dynamical system are converted to the integral items which can be easily calculated. Furthermore, the extended Melnikov function is employed to obtain the nonsmooth heteroclinic Melnikov chaotic threshold, which implies that the existence of the nonsmooth heteroclinic orbits may be due to the breaking of the nonsmooth heteroclinic loops under the perturbation of damping, external forcing and nonsmooth factors. It is worth pointing out that the form of the nonsmooth heteroclinic Melnikov function is different from the one of the nonsmooth homoclinic Melnikov function, which is quite different from the classical Melnikov theory.

Approximate controllability of second-order non-autonomous stochastic impulsive differential systems

This paper is concerned with existence, uniqueness, and approximate controllability of mild solutions to second-order non-autonomous stochastic impulsive differential systems. The impulsive differential systems may provide mathematical models to the phenomena having discontinuous jumps. The main results are carried out by using the generalized Banach contraction principle, evolution family, and stochastic analysis. Finally, an example is given to illustrate the results.

Finite-time stability of spring-mass system with unilateral impact constraints and frictions

In this paper, a spring-mass system with impacts and frictions is formulated by the impulsive differential system. An energy-like Lyapunov function and an auxiliary step function are constructed to analyze the finite-time stability of such impact system with a time-varying external force and sliding friction as well as air resistance. We establish the sufficient conditions of finite-time stability for three cases of the spring-mass system, and present numerical simulations for each case to verify the validity of the theoretical results.

Complex systems with impulsive effects and logical dynamics: A brief overview

In the past decades, complex systems with impulsive effects and logical dynamics have received much attention in both the natural and social sciences. This historical survey briefly introduces relevant studies on impulsive differential systems (IDSs) and logical networks (LNs), respectively. To begin with, we investigate five aspects of IDSs containing fundamental theory, Lyapunov stability, input-to-state stability, hybrid impulses and delay-dependent impulses. Next, we compactly summarize the research status of some problems of LNs including controllability, stability and stabilization, observability and current research. Moreover, some significant applications of proposed results are illustrated. Finally, based on this overview, we further discuss some future work on complex systems with impulsive effects and logical dynamics.

Exponential stability of non-linear stochastic delay differential system with generalized delay-dependent impulsive points

Abstract This paper is concerned with a non-linear stochastic delay differential system with delay-dependent impulsive perturbations. In this work, the size of the jump is defined as a general non-linear delay-dependent state variable and the solution of the impulsive stochastic delay differential system corresponding to the system without impulsive perturbations is given. This work is based on the relation between the solution of the equivalent model of stochastic delay differential system without impulses corresponding to the solution of the system with impulses. Then the conditions of the exponential stability of the proposed impulsive system are obtained by deriving stability criteria of the corresponding system without impulses. The numerical approximation for the stochastic delay system without impulses is developed using the Runge-Kutta-Maruyama method and it is suitably applied for the corresponding impulsive system. Finally, the obtained theoretical results are illustrated graphically for a stochastic delay system with impulses.

Nontrivial solutions for impulsive fractional differential systems through variational methods

Fractional differential equations (FDEs) as a generalization of ordinary differential equations and integration to arbitrary noninteger orders have gained importance due to their numerous applications in many fields of science and engineering. Indeed, there are a large number of phenomena, including fluid flow, diffusive transport akin to diffusion, rheology, probability, and electrical networks, that are modeled by different equations involving fractional order derivatives. This paper deals with multiplicity results of solutions for a class of impulsive fractional differential systems. The approach is based on variational methods and critical point theory. Indeed, we establish existence results for our system under some algebraic conditions on the nonlinear part with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear and the impulsive terms. Moreover by combining two algebraic conditions on the nonlinear term, which guarantee the existence of two weak solutions, applying the mountain pass theorem, we establish the existence of third weak solution for our system.

Impulsive Stability of Stochastic Functional Differential Systems Driven by G-Brownian Motion

This paper is concerned with the p-th moment exponential stability and quasi sure exponential stability of impulsive stochastic functional differential systems driven by G-Brownian motion (IGSFDSs). By using G-Lyapunov method, several stability theorems of IGSFDSs are obtained. These new results are employed to impulsive stochastic delayed differential systems driven by G-motion (IGSDDEs). In addition, delay-dependent method is developed to investigate the stability of IGSDDSs by constructing the G-Lyapunov–Krasovkii functional. Finally, an example is given to demonstrate the effectiveness of the obtained results.

Approximate controllability of semilinear impulsive functional differential systems with non-local conditions

AbstractThis paper is concerned with the approximate controllability of semilinear impulsive functional differential systems in Hilbert spaces with non-local conditions. We establish sufficient conditions for approximate controllability of such systems via resolvent operator and Schauder’s fixed point theorem. An application involving the impulse effect associated with delay and non-local conditions is presented to verify our claimed results.

Affine-Periodic Solutions for Impulsive Differential Systems

The affine-periodicity is a new periodic concept which has been founded in recent years. In this paper, we will discuss the existence of affine-periodic solutions for impulsive differential systems. Several existence theorems are proved for dissipative impulsive (functional) differential systems. Some applications are also given by combining Lyapunov’s methods.

Impulsive Synchronization of Fractional-Order Chaotic Systems With Actuator Saturation and Control Gain Error

This paper investigates the impulsive synchronization scheme of fractional-order chaotic systems with actuator saturation and control gain error. Based on the theory of fractional order system and impulsive differential system, discontinuous Lyapunov stability and matrix inequality approach, some new sufficient conditions are derived to guarantee the impulsive synchronization of a general class of fractional order chaotic systems. It is worth mentioning that the actuator saturation and control gain error are discussed simultaneously, which is more rigorous and practical in real systems. Finally, some simulation results verify the correctness and effectiveness of the theoretical results.

Leader-Following Consensus of Stochastic Perturbed Multi-Agent Systems via Variable Impulsive Control and Comparison System Method

In this paper, we explore the consensus control scheme of stochastic perturbed nonlinear multi-agent systems with impulsive protocol and comparison system method. The effective variable impulsive consensus method is used to remove the restriction of fixed impulsive instants, which is more reliable and flexible in practical applications. From the theory of impulsive differential system, comparison system and stochastic differential system, some sufficient consensus conditions are derived and the relation between the system parameters and impulsive time window is analyzed extensively. The effectiveness of the proposed control method is confirmed by the numerical simulations finally.

A novel Lyapunov theorem on finite/fixed-time stability of discontinuous impulsive systems.

This paper deals with the Finite/Fixed-Time Stability (FTS) problem of the discontinuous impulsive differential equation. Under the framework on differential inclusion, this problem can be transformed into the FTS problem of impulsive differential inclusion. A uniform criterion on FTS of nonlinear discontinuous impulsive differential systems with pre-given finite impulse instances is established, which is effective for both stabilizing impulses and destabilizing impulses. During this process, we propose an improved Lyapunov method, where the derivative of the Lyapunov Function (LF) may not exist in some instances. Moreover, the upper-bound estimation for the derivative of LF is allowed to be a time-varying function and takes both positive and negative values. Finally, the proposed criterion is supported by two numerical examples.

Optimal Impulsive Control With Application to Antiangiogenic Tumor Therapy

An optimal control algorithm is proposed for impulsive differential systems, i.e., systems evolving according to ordinary differential equations between any two control actions, occurring impulsively at discrete-time instants. Measurements are as well acquired at discrete-time instants. A model-based control law is conceived for medical and healthcare frameworks and, indeed, is applied to synthesize a feedback antiangiogenic tumor therapy. To cope with unavailable or temporally sparse measurements, the control law benefits a state observer properly designed for continuous-discrete systems by suitably exploiting recent results on observers for time-delay systems. The closed-loop algorithm is validated by building up an exhaustive simulation campaign on a population of virtual subjects, each sampled from a multivariate Gaussian distribution whose mean and covariance matrix is identified from experimental data taken from the literature. In silico results are encouraging and pave the way to further clinical verifications.

Optimal controllability of non-instantaneous impulsive partial stochastic differential systems with fractional sectorial operators

In this paper, we consider a new class of non-instantaneous impulsive partial stochastic differential systems with fractional sectorial operators in separable Hilbert spaces. Using the fractional calculus, the measure of noncompactness, properties of sectorial operators and fixed point theorems, we establish the optimal controllability to the these control systems without assuming the operators is compact. Finally, an example is provided to illustrate the obtained theory.

Some Improved Razumikhin Stability Criteria for Impulsive Stochastic Delay Differential Systems

This paper is devoted to study the Razumikhin stability theorem for a class of impulsive stochastic delay differential systems. By developing a new lemma, stochastic analysis technique, and Razumikhin approach, several novel criteria of the $p$ th moment exponential stability are derived for the related systems. The key feature of the criteria is that time-derivatives of the Razumikhin functions are allowed to be indefinite, which loosens the constraints of the existing results greatly. Finally, two examples are given to illustrate the usefulness and significance of the theoretical results.

Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects

This manuscript prospects the controllability analysis of non-instantaneous impulsive Volterra type fractional differential systems with state delay. By enroling an appropriate Grammian matrix with the assistance of Laplace transform, the conditions to obtain the necessary and sufficiency for the controllability of non-instantaneous impulsive Volterra-type fractional differential equations are derived using algebraic approach and Cayley-Hamilton theorem. A distinctive approach presents in the manuscript, i have taken non-instantaneous impulses into the fractional order dynamical system with state delay and studied the controllability analysis, since this not exists in the available source of literature. Inclusively, i have provided two illustrative examples with the existence of non-instantaneous impulse into the fractional dynamical system. So this demonstrates the validity and efficacy of our obtained criteria of the main section.

Adaptive odd impulsive consensus of multi-agent systems via comparison system method

This paper investigates the adaptive consensus problem of multi-agent nonlinear systems with odd impulsive control sequence. Based on the theory of impulsive differential system and Lyapunov stability, some new comparison system method-based sufficient criteria are derived to achieve the consensus of multi-agent nonlinear systems with uncertain parameters. Different from the existing methods of fixed impulsive consensus of multi-agent systems, the proposed impulsive control protocol with odd impulse sequence is more effective and flexible in real systems. Finally, the simulation is given to illustrate the effectiveness and validity of the theoretical results.

Controllability criteria of fractional differential dynamical systems with non-instantaneous impulses

AbstractThis manuscript prospects the controllability criteria of non-instantaneous impulsive Volterra type fractional differential systems. By enroling an appropriate Gramian matrix that is often defined by the Mittag-Leffler function and with the assistance of Laplace transform, the necessary and sufficiency conditions for the controllability of non-instantaneous impulsive Volterra-type fractional differential equations are derived by using algebraic approach and Cayley–Hamilton theorem. An important feature present in our paper is that we have taken non-instantaneous impulses into the fractional order dynamical system and studied the controllability analysis, since this do not exist in the available source of literature. Inclusively, we have provided two illustrative examples with the existence of non-instantaneous impulse into the fractional dynamical system. So this demonstrates the validity and efficacy of our obtained criteria of the main section.