Nonlinear Impulsive Control Research Articles

Existence of a mild solution and approximate controllability for fractional random integro-differential inclusions with non-instantaneous impulses

This paper investigates the existence and approximate controllability (ACA) of fractional neutral-type stochastic differential inclusions (NTSDIs) characterized by non-instantaneous impulses within a separable Hilbert space (HS) framework. Employing the Atangana–Baleanu–Caputo (ABC) derivative, we transform the system into an equivalent fixed-point (FP) problem through an integral operator. Subsequently, the Bohnenblust–Karlin FP theorem is leveraged to establish existence results. By assuming ACA of the corresponding linear system, we derive sufficient conditions for the ACA of the nonlinear stochastic impulsive control system. Our analysis relies on concepts from stochastic analysis, fractional calculus, FP theory, semigroup theory, and the theory of multivalued maps (MVMs). The theoretical findings are illustrated through a concrete example.

New exploration on approximate controllability of fractional neutral‐type delay stochastic differential inclusions with non‐instantaneous impulse

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper.

Backward bifurcation of a plant virus dynamics model with nonlinear continuous and impulsive control.

Roguing and elimination of vectors are the most commonly seen biological control strategies regarding the spread of plant viruses. It is practically significant to establish the mathematical models of plant virus transmission and regard the effect of removing infected plants as well as eliminating vector strategies on plant virus eradication. We proposed the mathematical models of plant virus transmission with nonlinear continuous and pulse removal of infected plants and vectors. In terms of the nonlinear continuous control strategy, the threshold values of the existence and stability of multiple equilibria have been provided. Moreover, the conditions for the occurrence of backward bifurcation were also provided. Regarding the nonlinear impulsive control strategy, the stability of the disease-free periodic solution and the threshold of the persistence of the disease were given. With the application of the fixed point theory, the conditions for the existence of forward and backward bifurcations of the model were presented. Our results demonstrated that there was a backward bifurcation phenomenon in continuous systems, and there was also a backward bifurcation phenomenon in impulsive control systems. Moreover, we found that removing healthy plants increased the threshold $ R_{1}. $ Finally, numerical simulation was employed to verify our conclusions.

Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response

<abstract><p>A good pulse control strategy should depend on the numbers of pests and natural enemies as determined via an integrated pest control strategy. Taking this into consideration, here, a nonlinear impulsive predator-prey model with improved Leslie-Gower and Beddington-DeAngelis functional response terms is qualitatively analyzed. The existence of a periodic solution for pest eradication has been obtained and the critical condition of global asymptotic stability has been established by using the impulsive differential equation Floquet theory. Furthermore, the conditions for the lasting survival of the system has been proved by applying a comparison theorem for differential equations. Additionally, a stable positive periodic solution has been obtained by applying bifurcation theory. To understand how nonlinear pulses affect the dynamic behavior of a system, MATLAB was used to conduct numerical simulations to show that the model has very complex dynamical behavior.</p></abstract>

Nonlinear Impulsive Control Design for Biologically Grounded NSM Tumor Growth Model Using Exact Linearized Mapping

This paper investigates the problem of personalized chemotherapy scheduling using a system-modeling approach based on a nonlinear impulsive control technique. In particular, this work deals with the so-called Norton-Simon-Massagué (NSM) model, a minimal biologically grounded model for controlled tumor growth, which has been accepted in clinical oncology for breast cancer treatment. In this paper, we continue this line of research, to advance the concept of closed-loop control a step further towards widespread clinical use. In the present work, the problem of chemotherapy drug scheduling is approached using an impulsive control strategy combined with a state-space observer. The technical challenges of the nonlinearity of the controlled system and the impulsive nature of control input are handled via the utilization of a novel mapping that transforms the control problem into a “simplified” linear discrete-time control problem. Theoretical guarantees regarding the sign and invertibility of the proposed transformation are demonstrated. Then, an observer-based state feedback controller is proposed for chemotherapy drug scheduling as a proof-of-concept control strategy. Simulation results showing the performance of the designed closed-loop controlled chemotherapy schedules in reducing and eradicating tumor volume are provided using model's parameters derived using real tumor data and Doxirubicin drug.

Dynamics of plant disease models with continuous and pulse farming awareness control

<p style='text-indent:20px;'>How to develop the mathematical model of plant disease transmission and consider the effect of saturated farming awareness strategies on plant disease control is of great practical significance. To do this, the plant disease transmission models with continuous and pulse farming awareness control strategy have been proposed in the present paper. For the model with continuous control strategies, the threshold values for the existence and stability of multiple equilibria have been given, and the effect of farming awareness on disease persistence is discussed; For the model with nonlinear impulsive control tactics, the existence and stability of the plant disease-free periodic solutions are investigated and the threshold condition is given, further we get the conditions for the permanence of the system and obtain the sufficient conditions under which the positive periodic solution exists by bifurcation theory. Numerical simulations reveal that people's farming awareness, especially global awareness, plays an important role in controlling and eradicating the plant disease. Moreover, the nonlinear impulsive control could result in the complex dynamic behavior, including period doubling, chaos and multiple attractors, which makes it difficult to design a successful plant disease control strategy.</p>

Non-smooth dynamics of a SIR model with nonlinear state-dependent impulsive control.

The classic SIR model is often used to evaluate the effectiveness of controlling infectious diseases. Moreover, when adopting strategies such as isolation and vaccination based on changes in the size of susceptible populations and other states, it is necessary to develop a non-smooth SIR infectious disease model. To do this, we first add a non-linear term to the classical SIR model to describe the impact of limited medical resources or treatment capacity on infectious disease transmission, and then involve the state-dependent impulsive feedback control, which is determined by the convex combinations of the size of the susceptible population and its growth rates, into the model. Further, the analytical methods have been developed to address the existence of non-trivial periodic solutions, the existence and stability of a disease-free periodic solution (DFPS) and its bifurcation. Based on the properties of the established Poincaré map, we conclude that DFPS exists, which is stable under certain conditions. In particular, we show that the non-trivial order-1 periodic solutions may exist and a non-trivial order-$ k $ ($ k\geq 1 $) periodic solution in some special cases may not exist. Moreover, the transcritical bifurcations around the DFPS with respect to the parameters $ p $ and $ AT $ have been investigated by employing the bifurcation theorems of discrete maps.

On stability of nonzero set-point for nonlinear impulsive control systems

The interest in non-linear impulsive systems (NIS) has been growing due to their impact on application problems such as disease treatments (diabetes, HIV, influenza, COVID-19, among many others), where the control action (drug administration) is given by short-duration pulses followed by time periods of null values. Within this framework, the concept of equilibrium needs to be extended (redefined) to allow the system to keep orbiting (between two consecutive pulses) in some state-space sets out of the origin, according to the usual objectives of most real applications. Although such sets can be characterized by means of a discrete-time system obtained by sampling the NIS at the impulsive times, no agreement has been reached on their asymptotic stability (AS) under optimizing control strategies. This paper studies the asymptotic stability of control equilibrium orbits for NIS, based on the underlying discrete-time system, in order to establish the conditions under which the AS for the latter leads to the AS for the former. Furthermore, based on the latter AS characterization, an impulsive Model Predictive Control (i-MPC) that feasibly stabilizes the non-linear impulsive system is presented. The biomedical problems of intravenous bolus administration of Lithium and antiretrovirals administration for HIV treatments are considered as simulation examples to demonstrate the controller performance

On Controllability of Impulsive Fractional Integro-differential Nonlocal System with State

This paper is concerned with the controllability of a nonlinear impulsive fractional integro-differential nonlocal control system with state-dependent delay in a Banach space. At first, we introduce a mild solution for the control system by using fractional calculus and probability density function. Under sufficient conditions, the results are obtained by means of semigroup theory and the Krasnoselskii fixed point theorem. Finally, an example is given to illustrate the main results.

Offset-Free Nonzero Tracking for Nonlinear Impulsive Systems With Application to Biomedical Processes

Impulsive control systems have shown strong potential to represent and address challenging problems, especially in the biomedical field. In recent research, these problems have been tackled with advances in linear impulsive control systems. However, many biomedical applications are better described by nonlinear impulsive models, and therefore, it is necessary to develop analysis tools and control strategies in this context. In the literature, the main properties of nonlinear impulsive control systems have been fully understood, but there is no major development of control strategies. Particularly, there is no substantiation of model predictive control (MPC) strategies maintaining convexity of the optimization problem and closed-loop stability, and there is no control strategy to reduce the offset problem when there are parameter variations, which is a common situation in biological processes. Therefore, the main novelties of this paper are: (i) an MPC formulation extended to nonlinear impulsive systems that addresses non-zero tracking, (ii) the sufficient and necessary conditions to guarantee the stability of the closed-loop system at an equilibrium target, (iii) a comprehensive description of an offset-free MPC to handle low to moderate plant-model mismatches, (iv) the conditions to guarantee offset-free control. Finally, the MPC and offset-free MPC are tested to address the drug administration problem in two biomedical applications: oncolytic virus therapy, to regulate tumor dynamics using doses of oncolytic, and type 1 diabetes treatment, to regulate glycemia using insulin injections. Satisfactory results were obtained in simulation scenarios including parameter variations in nonlinear models that represent the corresponding dynamics.

Fuzzy Coefficient of Impulsive Intensity in a Nonlinear Impulsive Control System

Fuzzy Coefficient of Impulsive Intensity in a Nonlinear Impulsive Control System

Dynamics of a density-dependent predator-prey biological system with nonlinear impulsive control.

Spraying insecticides and releasing natural enemies are two commonly used methods in the integrated pest management strategy. With the rapid development of biotechnology, more and more realistic factors have been considered in the establishment and implementation of the integrated pest management models, such as the limited resources, the mutual restriction between pests and natural enemies, and the monitoring data of agricultural insects. Given these realities, we have proposed a pest-natural enemy integrated management system, which is a nonlinear state-dependent feedback control model. Besides the anti-predator behavior of the pests to the natural enemies is considered, the density dependent killing rate of pests and releasing amount of natural enemies are also introduced into the system. We address the impulsive sets and phase sets of the system in different cases, and the analytic expression of the Poincaré map which is defined in the phase set was investigated. Further we analyze the existence, uniqueness, global stability of order-1 periodic solution. In addition, the existence of periodic solution of order-k (k≥2) is discussed. The theoretical analyses developed here not only show the relationship between the economic threshold and the other key factors related to pest control, but also reveal the complex dynamical behavior induced by the nonlinear impulsive control strategies.

Complex Dynamics of Beddington–DeAngelis-Type Predator-Prey Model with Nonlinear Impulsive Control

According to resource limitation, a more realistic pest management is that the impulsive control actions should be adjusted according to the densities of both pest and natural enemy in the field, which result in nonlinear impulsive control. Therefore, we have proposed a Beddington–DeAngelis interference predator-prey model concerning integrated pest management with both density-dependent pest and natural enemy population. We find that the pest-eradication periodic solution is globally stable if the impulsive period is less than the critical value by Floquet theorem. The condition of permanent is established, and a stable positive periodic solution appears via a supercritical bifurcation by bifurcation theorem. Finally, in order to investigate the effects of those nonlinear control strategies on the successful pest control, the bifurcation diagrams showed that the model exists with very complex dynamics. Consequently, the resource limitation may result in pest outbreak in complex ways, which means that the pest control strategies should be carefully designed.

Stability Analysis for Nonlinear Impulsive Control System with Uncertainty Factors.

Considering the limitation of machine and technology, we study the stability for nonlinear impulsive control system with some uncertainty factors, such as the bounded gain error and the parameter uncertainty. A new sufficient condition for this system is established based on the generalized Cauchy–Schwarz inequality in this paper. Compared with some existing results, the proposed method is more practically applicable. The effectiveness of the proposed method is shown by a numerical example.

Existence-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal control

In this thesis, we investigate a kind of impulsive fractional order differential systems involving control terms. By using a class of φ-concave-convex mixed monotone operator fixed point theorem, we obtain a theorem on the existence and uniqueness of positive solutions for the impulsive fractional differential equation, and the optimal control problem of positive solutions is also studied. As applications, an example is offered to illustrate our main results.

Complex dynamics and coexistence of period-doubling and period-halving bifurcations in an integrated pest management model with nonlinear impulsive control

An expectation for optimal integrated pest management is that the instantaneous numbers of natural enemies released should depend on the densities of both pest and natural enemy in the field. For this, a generalised predator–prey model with nonlinear impulsive control tactics is proposed and its dynamics is investigated. The threshold conditions for the global stability of the pest-free periodic solution are obtained based on the Floquet theorem and analytic methods. Also, the sufficient conditions for permanence are given. Additionally, the problem of finding a nontrivial periodic solution is confirmed by showing the existence of a nontrivial fixed point of the model’s stroboscopic map determined by a time snapshot equal to the common impulsive period. In order to address the effects of nonlinear pulse control on the dynamics and success of pest control, a predator–prey model incorporating the Holling type II functional response function as an example is investigated. Finally, numerical simulations show that the proposed model has very complex dynamical behaviour, including period-doubling bifurcation, chaotic solutions, chaos crisis, period-halving bifurcations and periodic windows. Moreover, there exists an interesting phenomenon whereby period-doubling bifurcation and period-halving bifurcation always coexist when nonlinear impulsive controls are adopted, which makes the dynamical behaviour of the model more complicated, resulting in difficulties when designing successful pest control strategies.

Chaos suppression in speed control for permanent-magnet-synchronous-motor drive system

This paper considers chaos in a PI control system for the first time and presents an equivalent-input-disturbance (EID)-based control method to suppress a chaotic phenomenon. Since the chaos is caused by two nonlinear terms, two EID estimators are used to separately compensate for them, and thus to suppress any possible chaos. This paper devises a new way to analyze the stability of the EID-based PMSM drive system and provides a stability region based on the concept of local uniformly boundedness. Simulation results show that the method is effective to suppress chaos for the PI control system. A comparison of the method with nonlinear feedback control, backstepping control, and impulsive control shows the superiority of our method.

Asymptotic stability analysis via indefinite Lyapunov functions and design of nonlinear impulsive control systems

In this paper, we study asymptotic stability problem of nonlinear impulsive dynamic systems and design an impulsive controller for a chaotic system. We propose new sufficient conditions for asymptotic stability of the origin of nonlinear impulsive dynamic systems via indefinite Lyapunov functions. Indefinite Lyapunov functions may increase both during some continuous portion of the trajectory and at some impulses. We present two examples to demonstrate the effectiveness of our conclusions. Furthermore, based on the results, impulsive control is designed for a chaotic system.

Lyapunov Stability for Impulsive Systems via Event-Triggered Impulsive Control

In this article, we investigate the Lyapunov stability problem for impulsive systems via event-triggered impulsive control, where dynamical systems evolve according to continuous-time equations most of the time, but occasionally exhibit instantaneous jumps when impulsive events are triggered. We provide some Lyapunov-based sufficient conditions for uniform stability and globally asymptotical stability. Unlike normal time-triggered impulsive control, event-triggered impulsive control is triggered only when an event occurs. Thus our stability conditions rely greatly on the event-triggering mechanism given in terms of Lyapunov functions. Moreover, the Zeno behavior can be excluded in our results. Then, we apply the theoretical results to the nonlinear impulsive control system, where event-triggered impulsive control strategies are designed to achieve stability of the addressed system. Finally, two numerical examples and their simulations are provided to demonstrate the effectiveness of the proposed results.

Stability analysis for nonlinear impulsive optimal control problems

Stability analysis for nonlinear impulsive optimal control problems