A numerical method for an optimal control problem with minimum sensitivity on coefficient variation

In this paper, we consider a class of optimal control problem involving an impulsive systems in which some of its coefficients are subject to variation. We formulate this optimal control problem as a two-stage optimal control problem. We first formulate the optimal impulsive control problem with all its coefficients assigned to their nominal values. This becomes a standard optimal impulsive control problem and it can be solved by many existing optimal control computational techniques, such as the control parameterizations technique used in conjunction with the time scaling transform. The optimal control software package, MISER 3.3, is applicable. Then, we formulate the second optimal impulsive control problem, where the sensitivity of the variation of coefficients is minimized subject to an additional constraint indicating the allowable reduction in the optimal cost. The gradient formulae of the cost functional for the second optimal control problem are obtained. On this basis, a gradient-based computational method is established, and the optimal control software, MISER 3.3, can be applied. For illustration, two numerical examples are solved by using the proposed method.     

Impulsive generalized synchronization for a class of nonlinear discrete chaotic systems

The problem of impulsive generalized synchronization for a class of nonlinear discrete chaotic systems is investigated in this paper. Firstly the response system is constructed based on the impulsive control theory. Then by the asymptotic stability criteria of discrete systems with impulsive effects, some sufficient conditions for asymptotic H-synchronization between the drive system and response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

On optimal control problems of a class of impulsive switching systems with terminal states constraints

The global optimal control problem is proposed for a special class of hybrid dynamical systems, i.e. impulsive switching systems. Then the necessary condition of the above problem, the minimum principle, is given. Ekeland’s variational principle and the matrix cost functional structure expression are utilized in the process of the proof. Based on the main result, a special linear hybrid impulsive and switching system (HISS) is illustrated and the optimal control algorithm is presented. Moreover, the cases of pure impulsive systems and pure switched systems are included in this paper.     

Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations

In this paper, we study optimal relaxed controls and relaxation of nonlinear fractional impulsive evolution equations. Firstly, existence of piecewise continuous mild solutions for the original fractional impulsive control system is presented. Secondly, fractional impulsive relaxed control system is constructed by using a regular countably additive measure and making the original control system convexified. Thirdly, optimal relaxed controls and relaxation theorems are obtained. Finally, application to initial-boundary value problem of fractional impulsive parabolic control system is considered.     

Global exponential synchronization of nonlinear time-delay Lur’e systems via delayed impulsive control

This paper is concerned with the global exponential synchronization problem of two identical nonlinear time-delay Lur’e systems via delayed impulsive control. Some novel impulsive synchronization criteria are obtained by introducing a discontinuous Lyapunov function and by using the Lyapunov–Razumikhin technique, which are expressed in forms of linear matrix inequalities. The derived criteria reveal the effects of impulsive input delays and impulsive intervals on the stability of synchronization error systems. Then, sufficient conditions on the existence of a delayed impulsive controller are derived by employing these newly-obtained synchronization criteria. Additionally, some synchronization criteria for two identical time-delay Lur’e systems with impulsive effects are presented by using delayed continuous feedback control. The synchronization criteria via delayed continuous feedback control can deal with the case when the impulsive control strategy fails to synchronize two identical impulsive time-delay Lur’e systems. Three numerical examples are provided to illustrate the efficiency of the obtained results.     

Event-triggered impulsive control of lower-triangular large-scale nonlinear systems based on gain scaling technique

This paper investigates the event-triggered impulsive control problem for a class of large-scale nonlinear systems in lower-triangular form. Based on gain scaling technique and impulsive control theory, a novel decentralized event-triggered impulsive control strategy is first put forward by introducing a static scaling gain, where no control input exists between two consecutive triggering points. Moreover, when the large uncertainties exist in system nonlinearities, we further develop a new control strategy by introducing a time-varying scaling gain. It is proved that the proposed closed-loop control strategies exclude the Zeno behavior without sacrificing the global convergence of system states. Compared with the existing results, it is the first time to apply impulsive control to lower-triangular large-scale nonlinear systems, and the advantages of event-triggered impulsive control and gain scaling technique are subtly combined in the proposed control strategies. Finally, two simulation examples are used to demonstrate the effectiveness of the proposed schemes.     

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

Event-triggered and self-triggered impulsive control for two-time-scale systems

This paper studies the stability problem of two-time-scale system via event-triggered impulsive control and self-triggered impulsive control. The overall system is modeled with the hybrid formalism. Two Chang transformations are introduced to completely decouple the hybrid system states into flow set and jump set. A composite impulsive controller based on slow and fast system states is proposed, under which the slow and fast subsystems are simultaneously triggered by event-triggered and self-triggered mechanism, respectively. As a result, the stability conditions are derived for the system under event-triggered and self-triggered impulsive control, respectively. Furthermore, the theoretical result of self-triggered impulsive control is applied to the consensus of the interconnected two-time-scale systems. Finally, simulation examples and comparison study show the effectiveness of the proposed control strategies.     

On value functions for impulsive control of piecewise-deterministic processes

The paper deals with value functions for optimal stopping and impulsive control for piecewise-deterministic processes with discounted cost. The associated dynamic programming equations are variational and quasi-variational inequalities with integral and first-order differential terms The technique used is to approximate the value functions for an optimal stopping (impulsive control. switching control) problem for a piecewise-deterministic process by value functions for optimal stopping (impulsive control, switching control) problems for Feller piecewise-deterministic processes     

Impulsive controller design for exponential synchronization of chaotic neural networks with mixed delays

This paper considers the chaotic synchronization problem of neural networks with time-varying and distributed delays using impulsive control method. By utilizing the stability theory for impulsive functional differential equations, several impulsive control laws are derived to guarantee the exponential synchronization of neural networks with time-varying and distributed delays. It is shown that chaotic synchronization of the networks is heavily dependent on the designed impulsive controllers. Moreover, these conditions are expressed in terms of LMI and can be easily checked by MATLAB LMI toolbox. Finally, a numerical example and its simulation are given to show the effectiveness and advantage of the proposed control schemes.     

Exponential stabilization of nonlinear systems under saturated control involving impulse correction

This paper studies the problem of locally exponential stabilization (LES) of nonlinear systems under a class of hybrid control in the framework of actuator saturation, where the limitation of actuator saturation on both continuous state feedback control and impulsive control are fully considered. Based on impulsive control theory and differential inclusion approach, some sufficient conditions for LES are derived, where a novel set inclusion relation is proposed to handle the double saturation nonlinearities. Different from the existing results that each part of saturated hybrid control (SHC) is required to stabilize the system individually, our results relax the requirement by making full use of the correction effect of the impulse. Moreover, the maximum of estimation of domain of attraction is obtained by a convex optimal problem and corresponding algorithm. The result is applied to the robustness for a class of nonlinear systems. Finally, the validity of the results is shown by two examples and their simulations, where the synchronization problem of Chua’s oscillator is illustrated in the framework of actuator saturation.     

A new impulsive synchronization criterion for T–S fuzzy model and its applications

This paper focuses on the problem of impulsive synchronization of T–S fuzzy systems. A new synchronization criterion is derived for T–S fuzzy systems by utilizing the concept of average impulsive interval. The proposed impulsive control scheme has a simple control structure, and is theoretically and numerically proved to be less conservative than some existing results. The method is also illustrated by applying to Lorenz system, Rössler’s system as well as permanent magnet synchronous motors system.     

Feasibility of time-limited control of a competition system with impulsive harvest

A kind of time-limited control model on a competition system with impulsive harvest, described by impulsive differential equation with the initial and boundary value problem, is presented. The existence of solution of the model, corresponding to the feasibility of the short-term control, is discussed. By the comparison principle, the conditions under which the model has a solution are found by a series of the upper solutions, and the conditions under which the model has no solution are also given by a series of the lower solutions. Finally, the practical meanings of those conditions are explained. As an example, if other parameters are given, the times of the impulsive control is estimated and the theoretical results are verified by numerical simulations.     

On Risk-Sensitive Ergodic Impulsive Control of Markov Processes

Impulsive control of continuous-time Markov processes with risk- sensitive long-run average cost is considered. The most general impulsive control problem is studied under the restriction that impulses are in dyadic moments only. In a particular case of additive cost for impulses, the impulsive control problem is solved without restrictions on the moments of impulses. Accepted 30 April 2001. Online publication 29 August 2001.     

The Canonical Theory of the Impulse Process Optimality

The paper is devoted to the development of the canonical theory of the Hamilton–Jacobi optimality for nonlinear dynamical systems with controls of the vector measure type and with trajectories of bounded variation. Infinitesimal conditions of the strong and weak monotonicity of continuous Lyapunov-type functions with respect to the impulsive dynamical system are formulated. Necessary and sufficient conditions of the global optimality for the problem of the optimal impulsive control with general end restrictions are represented. The conditions include the sets of weak and strong monotone Lyapunov-type functions and are based on the reduction of the original problem of the optimal impulsive control a finite-dimensional optimization problem on an estimated set of connectable points.     

Variable impulsive consensus of nonlinear multi-agent systems

In this paper, the consensus problem for nonlinear multi-agent systems with variable impulsive control method is studied. In order to decrease the communication wastage, a novel distributed impulsive protocol is designed to achieve consensus. Compared with the common impulsive consensus method with fixed impulsive instants, the variable impulsive consensus method proposed in this paper is more flexible and reliable in practical application. Based on Lyapunov stability theory and some inequality techniques, several novel impulsive consensus conditions are obtained to realize the consensus of multi-agent systems. Finally, some necessary simulations are performed to validate the effectiveness of theoretical results.     

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.     

Hybrid output regulation for linear impulsive systems with aperiodic jumps: A discrete-time feedback controller design approach

This paper is concerned with the problem of hybrid output regulation for a class of linear impulsive systems with aperiodic jumps. Firstly, by leveraging time-dependent Lyapunov function technique and impulsive control theory, sufficient conditions for achieving output regulation are obtained in state feedback case. Then, the results are extended to error feedback case by constructing an impulsive observer. In this framework, two novel hybrid controllers are designed. Such controllers only need the discrete-time system state or error signal for feedback. The complete procedures for controller designs are also presented. Finally, two illustrative examples, including a numerical example and an LC circuit, are given to show the validity and applicability of the proposed control laws.     

Mathematical models of restoration and control of a single species with Allee effect

According to the initial density of a single species with Allee effect and corresponding management strategy, three kinds of mathematical models are presented to describe the evolutionary process of the species by impulsive differential equations. When the initial density of the species is larger than economic injury level (EIL) (or economical threshold, ET), impulsive harvest control is considered in a finite time to decrease the population of the species. The feasibility of the impulsive harvest control in a finite time is given by the existence of solution of the model with initial and boundary value problem. When the initial density of the species is less than EIL (or ET), the model with state feedback control is formulated according to the state of the species. The existence and stability of periodic solution of the model with state feedback control are discussed. When the initial density of the species is less than the Allee threshold and the species tends to extinction, the model with impulsive release at fixed moments is presented to study the restoration of the species. The conditions for the feasibility of periodic restoration of the species are given. Finally, some discussions are given.     

A maximum principle for smooth optimal impulsive control problems with multipoint state constraints

A nonlinear optimal impulsive control problem with trajectories of bounded variation subject to intermediate state constraints at a finite number on nonfixed instants of time is considered. Features of this problem are discussed from the viewpoint of the extension of the classical optimal control problem with the corresponding state constraints. A necessary optimality condition is formulated in the form of a smooth maximum principle; thorough comments are given, a short proof is presented, and examples are discussed.