On ergodic impulsive control problems

The impulsive control with long run average cost for Feller-Markov processes with quasicompact semigroups is studied.     

Lyapunov stability for impulsive control affine systems

This article studies several notions of Lyapunov stability for impulsive control affine systems in the setting of nonautonomous dynamical systems. It presents some relations between the stability of an impulsive control affine system and the stability of its adjacent control system. Stability of compact sets and their components are specially investigated. Lyapunov functionals are employed to characterize each type of stability of closed sets.     

Stability analysis of impulsive control systems

This paper studies impulsive control systems. Several stability criteria are established by employing the method of Lyapunov functions. These criteria may be used for impulsive feedback control design. As an application, impulsive control of the Lorenz chaotic system is discussed. Numerical experiments are carried out for the control of the Lorenz system. It is shown that small and frequent impulses need to be used in order to stabilize the Lorenz system.     

Causal modeling in impulsive systems: A new rigorous non-standard analysis approach

A new approach for modeling nonlinear impulsive system is suggested based on nonstandard analysis. Basic properties of the hyperreals in nonstandard analysis are revisited. Depending on the convergence rate of infinitesimals in hyperreals, a new extended real space is proposed, which extends the one dimensional real line to a countably infinite dimensional extended real space. Generalized functions are defined via a sequential approach on the extended space, which yields a class of Heaviside functions and singular functions. By using the extended functions, a causal way for characterizing jumps in discontinuous system follows. We illustrate the usefulness of the theoretical development by analyzing three simple cases of impulsive affine system: (1) scalar case, (2) multi-dimensional case, and (3) one dimensional horizontal bouncing ball. The results suggest not only the existence of such infinitesimal models within the jump but also how to detour the equilibrium point while connecting the discontinuous state at the impact.     

A study on time-limited control of single-pest with stage-structure

Two kinds of time-limited pest control models of single-pest with stage-structure, which can be described by the boundary value problem of ordinary differential equation and impulsive differential equation, are presented according to the ways of artificial control (continuous control and impulsive control). The conditions under which the corresponding model has a solution are given. If the model has a solution, the corresponding aim of pest control can be achieved. The theoretical results show that both the mature and the immature pest should be controlled synchronously, otherwise the aims of pest control can not be achieved in a finite time. Finally, some discussions and numerical simulations show that the impulsive control is more practical than the continuous control.     

Impulsive consensus in directed networks of identical nonlinear oscillators with switching topologies

In this paper, we investigate the problem of impulsive consensus of multi-agent systems, where each agent can be modeled as an identical nonlinear oscillator. Firstly, an impulsive control protocol is designed for directed networks with switching topologies based on the local information of agents. Then sufficient conditions are given to guarantee the consensus of the networked nonlinear oscillators. How to select the discrete instants and impulsive constants is also discussed. Numerical simulations show the effectiveness of our theoretical results.     

The study of a predator–prey system with group defense and impulsive control strategy

A predator–prey system with group defense and impulsive control strategy is established. By using Floquet theorem and small amplitude perturbation skills, a locally asymptotically stable prey-eradication periodic solution is obtained when the impulsive period is less than some critical value. Otherwise, if the impulsive period is larger than the critical value, the system is permanent. By using bifurcation theory, we show the existence and stability of positive periodic solution when the pest-eradication lost its stability. Further, numerical examples show that the system considered has more complicated dynamics, such as: (1) quasi-periodic oscillating, (2) period-doubling bifurcation, (3) period-halving bifurcation, (4) non-unique dynamics (meaning that several attractors coexist), (5) attractor crisis, etc. Finally, the biological implications of the results and the impulsive control strategy are discussed.     

Impulsive control strategy of a pest management SI model with nonlinear incidence rate

In this work, we consider a pest management SI model with impulsive release of infective pests and spraying pesticides. We prove that all solutions of the investigated system are uniformly ultimately bounded and the pest-extinction periodic solution is globally asymptotically stable when some condition is satisfied. We also obtain the permanent condition of the system. It is concluded that the approach of combining impulsive release of infective pests with impulsive spraying pesticides provides reliable tactic basis for the practical pest management.     

Self-triggered impulsive control of nonlinear time delay systems: Application to chemotherapeutic dose-regimen design

This paper proposes a self-triggered impulsive control for nonlinear time-delay systems, where the time instant of the next impulsive input is calculated based on the last measurement and the values of the systems’ parameters. Contrary to event-triggered scheme where the actuation is discrete but a continuous monitoring of the system’s states is necessary, in the self-triggered approach, both sampling and actuation are performed at distinct moments of time. Utilizing Lyapunov–Razumikhin method and by upper-bounding the system’s trajectory, the global asymptotic stability of the system’s equilibrium is verified when the rate of change in the Lyapunov function is exponential. In the general case, the global ultimate boundedness of the system’s trajectory is shown where the ultimate bound can be set arbitrarily small. As an application, for the first time, the problem of dose regimen design is formulated in the sampled data framework. Then, based on the obtained theoretical results, the appropriate regimen is suggested. In particular, time-triggered and self-triggered therapy protocols for docetaxel, a phase specific chemotherapeutic drug which is administered intravenously, are proposed. Clinical constraints such as maximum tolerated dose, discontinuous drug administration, and intermittent measurements are met in the proposed therapy protocols. According to in-silico results, both proposed self-triggered and time-triggered dose regimens outperform the traditional weekly fixed dose administration. Finally, the robustness of the proposed schemes to parameter uncertainties is evaluated through an extensive set of simulations.     

Graph completions for impulsive feedback controls

The paper considers impulsive control systems, where the evolution equation depends linearly on the time derivatives of the control function. The basic theory of “graph completions” is here extended to control functions in feedback form. In the case where the control u=u(t,x)$u=u(t,x)$ is piecewise smooth, with a jump along a hypersurface Σ in the t–x space, results are proved on the existence and uniqueness of solutions, and on their approximation by means of smooth feedbacks. The paper is concluded with a couple of examples, concerning the feedback control of mechanical systems by means of active constraints.     

Impulsive stabilization of complex-valued stochastic complex networks via periodic self-triggered intermittent control

The aim of this article is to research the stabilization issue of complex-valued stochastic Markovian switching complex networks with time delays and time-varying multi-links (CSMCTM) via periodic self-triggered intermittent impulsive control (PSIIC). Thereinto, PSIIC is designed for the first time by combining intermittent impulsive control with periodic self-triggered control for intermittent control. It is worth emphasizing that the triggered protocol is designed to be more flexible, and easier to implement than previously reported triggered protocol. Then, by means of impulsive control, intermittent control, event-driven control theory and stability analysis, a stabilization criterion of CSMCTM in the sense of exponential stability in mean square is obtained. Whereafter, the stability of a class of complex-valued inertial neural networks is researched as a practical application of our theoretical results. Ultimately, a numerical example gives a corresponding verification.     

Passive and impulsive synchronization of a new four-dimensional chaotic system

This paper studies the synchronization problem for a new chaotic four-dimensional system presented by Qi et al. Two different methods, the passive control method and the impulsive control method, are used to control the synchronization of the four-dimensional chaotic system. Numerical simulations show the effectiveness of the two different methods.     

Impulsive consensus problem of second-order multi-agent systems with switching topologies

The paper proposes an impulsive consensus protocol to solve the consensus problem of the second-order multi-agent systems with fixed and switching topologies. Some sufficient conditions are obtained for the states of follower agents converging to the state of leader asymptotically. Two numerical simulations are also given to verify the effectiveness of the theoretical analysis.     

Target problems under state constraints for nonlinear controlled impulsive systems

We study the problem of reaching a target without leaving a prescribed constraint set for a dynamical system described by a controlled differential equation and a controlled instantaneous reset function. We characterize all initial conditions from which the objective can be reached. Then we characterize the value function associated with the optimal reaching time problem.     

Solvability and Optimal Controls of Fractional Impulsive Stochastic Evolution Equations with Nonlocal Conditions

This paper deals with the solvability and optimal controls of a class of impulsive fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. Firstly, the existence and uniqueness of mild solutions for the considered system are investigated. Then, we derive the existence conditions of optimal pairs to the control systems. In the end, an example is presented to illustrate the effectiveness of our abstract results.     

Hybrid event-triggered and impulsive control for time-delay systems

In this paper, we study the problem of hybrid event-triggered control for a class of nonlinear time-delay systems. Using a Razumikhin-type input-to-state stability result for time-delay systems, we design an event-triggered control algorithm to stabilize the given time-delay system. In order to exclude Zeno behavior, we combine the impulsive control mechanism with our event-triggered strategy. In this sense, the proposed algorithm is a hybrid impulsive and event-triggered strategy. Sufficient conditions for the stabilization of the nonlinear systems with time delay are obtained by using Lyapunov method and Razumikhin technique. Numerical simulations are provided to show the effectiveness of our theoretical results.     

On the compactness method in general ergodic impulsive control of markov processes

In this paper the impulsive control of Feller Markov processes on compact state space with long run average cost criterion is studied. Under the assumption of compactness of the resolvent operator the optimal strategies corresponding to general cost for impulses are constructed. Also the case of purely jump Markov processes is considered     

害虫治理的病毒感染模型

研究了食饵受病毒感染且捕食者具有Beddington-DeAngelis功能性反应的生态流行病模型,此模型考虑的是脉冲释放病毒颗粒和自然天敌. 利用Floquet乘子理论、小振幅扰动技巧和比较定理证明了害虫根除周期解的全局渐近稳定性以及系统持续生存的充分条件.结论为现实的害虫管理提供了有效的策略依据.