State-dependent Impulses Research Articles

Second–order discontinuous ODEs and billiard problems

We present an existence principle for boundary value problems involving discontinuous ordinary differential equations of the second order using the Krasovskii regularization technique. Especially we obtain sufficient conditions of transversality type for Krasovskii solutions to be also Carathéodory solutions of the original problem. This result is applied on a certain billiard problem, which can be thought as an ordinary differential equation with state-dependent impulses that is equivalent to certain discontinuous differential equation. In particular, we obtain new existence and multiplicity results for Dirichlet problems in billiard spaces with time-varying boundaries.

Pinning Controller Design for Set Reachability of State-Dependent Impulsive Boolean Networks.

Considered the stimulation of tumor necrosis factor as an impulsive control, an apoptosis network is modeled as a state-dependent impulsive Boolean network (SDIBN). Making cell death normally means driving the trajectory of an apoptosis network out of states that indicate cell survival. To achieve the goal, this article focuses on the pinning controller design for set reachability of SDIBNs. To begin with, the definitions of reachability and set reachability are introduced, and their relation is illustrated. For judging whether the trajectory of an SDIBN leaves undesirable states, a necessary and sufficient condition is presented according to the criteria for the set reachability. In addition, a series of algorithms is provided to find all possible sets of pinning nodes for the set reachability. Note that attractors containing in all undesirable states are studied to make SDIBNs set reachable via controlling the smallest states. For the purpose of determining pinning nodes for one-step set reachability, the Hamming distance is presented under scalar forms of states. Pinning nodes with the smallest cardinality for the set reachability are derived by deleting some redundant nodes. Compared with the existing results, the state feedback gain can be obtained without solving logical matrix equations. The computation complexity of the proposed approach is lower than that of the existing methods. Moreover, the method of designing pinning controllers is used to discuss apoptosis networks. The experimental result shows that apoptosis networks depart from undesirable states by controlling only one node.

The only limit cycle that appears in damped harmonic oscillators affected by state-dependent impulses

This study is devoted to planar hybrid dynamical systems with state-dependent impulses to investigate phenomena related to sudden changes in the velocity of an object. In particular, the effect of the state-dependent impulses on the appearance of limit cycles is addressed, which has rarely been studied to date. When a toddler is placed on a swing and pushed gently from behind, the discontinuous changes in the direction and speed of movement of both the toddler and the swing can be described using such a hybrid dynamical system. The main theorem presented in this study reveals that, regardless of the amount of state-dependent impulses, they yield only one stable limit cycle within the region that is identified by the considered system. Moreover, state-dependent impulses lead to destabilization of the equilibrium point. In the proof of the main theorem, a Poincaré mapping is derived by examining the behavior of positive semi-orbits, and the characteristics of the sequence that is obtained from the Poincaré mapping are investigated in detail. Several simulations are performed to demonstrate the main theorem.

Long-time behavior for impulsive generalized semiflows

We propose new criteria, which are more general than those previously known in the literature, to guarantee the existence of global attractors for problems with state-dependent impulses. Our result on the existence of global attractors is proved for the case where the solutions on the underlying semiflow can possibly be nonunique, and it is even new for the case of solution uniqueness. Additionally, we provide collective versions of the proposed criteria, under which, as we prove, the global attractors exhibit upper-semicontinuity upon perturbation of the problem. The theory is illustrated through several examples of ODEs and PDEs.

On the jump control problem for boundary-value problems with state-dependent impulses

UDC 517.9 We show how an appropriate parametrization technique and special successive approximations can help to control unknown jumps in the case of nonlinear boundary-value problems with state-dependent impulses. The practical application of the technique is shown on a numerical example.

Optimal control of a nonlinear state-dependent impulsive system in fed-batch process

Optimal control technique is crucial to improve the yield of microbial fermentation production. In this paper, we propose a nonlinear control system with state-dependent impulses, where the impulsive volume of feeding glycerol and the critical concentration of glycerol for occurring impulse are the control variables, to formulate 1,3-propanediol (1,3-PD) fed-batch production process. We also discuss a quantity of important properties for this control system. Then, we analyze the sensitivity of system state with respect to the kinetic parameters. We further propose a constrained optimal control model governed by the control system with state-dependent impulses. The existence of the optimal impulsive controls is established. For solving this problem, we utilize an exact penalty method to transform the problem into an optimization problem with only box constraints. Moreover, an improved differential evolution method is developed to seek the optimal impulsive strategy. Finally, numerical simulation results demonstrate that, by using the optimal impulsive strategies, final 1,3-PD concentration is considerably increased under the nominal parameter values and disturbances of kinetic parameters have significant effects on the optimal final 1,3-PD yield.

Input-to-State Stability of Impulsive Systems via Event-Triggered Impulsive Control.

This article investigates input-to-state stability (ISS) and integral ISS (iISS) of nonlinear impulsive systems based on the event-triggered impulsive control (ETIC) strategy, where the impulse sequence is generated by some predesigned event conditions. Unlike traditional event-triggered control, ETIC means that the controller is activated only when some state-dependent event conditions are triggered and moreover, there is not any control transmission between two consecutive triggered impulse instants. Event-triggered impulses are usually regarded as a class of state-dependent impulses, where the event-triggered mechanism (ETM) is an impulse generator. By using the ETIC strategy, some Lyapunov-based criteria are established, which can effectively avoid infinitely fast triggering behavior and guarantee ISS/iISS of nonlinear impulsive systems. Then, the theoretical results are applied to nonlinear system, where a class of ETMs and impulsive control gain are derived with the help of LMIs. Finally, two numerical examples are presented to illustrate the validity of our control strategies.

Qualitative behavior of solutions of Liénard-type systems with state-dependent impulses

State-dependent impulsive differential equations can be used to describe phenomena in which the velocity of an object suddenly changes when the object enters a predetermined state. This study examines the effect of state-dependent impulses on the rotation of all orbits of the Liénard system, which plays an important role in many research areas, including electrical circuits, diverse engineering fields, economics, ecology, and physiology. Determining whether all orbits of the Liénard system other than the origin, which is the only fixed point, rotate around the origin, is the basis of other properties of the orbit and has become a significant research subject; however, very few studies have examined the effects of state-dependent impulses on this behavior, and the main theorem presented in this paper addresses this. This rotation problem is reduced to establishing whether all orbits intersect the vertical isocline, which is discussed in detail. To facilitate the understanding of the proof of the main theorem, an overview is presented before providing the actual proof. The main theorem and some lemmas are proved using phase plane analysis. The application of the main theorem to Euler’s equations is also described.

Limit cycles of a class of Liénard systems derived from state-dependent impulses

Discontinuous phenomena, in which objects may behave continuously and sometimes discretely are not only found in nature and under laboratory conditions but also in simple, familiar contexts. For example, this phenomenon is skillfully incorporated into the internal structure of mechanical wristwatches. Unless an extremely small amount of state-dependent impulse is applied intermittently, the reciprocating rotational movement of the balance and hairspring, which is the heart of the mechanical wristwatch, cannot be maintained. The small amount of state-dependent impulse, which is often overlooked, can make a significant difference; however, very few studies have examined this subject. This study assumes the underlying cause of discontinuous behaviors as impulses generated when an object reaches a particular state, assuming that the continuous behavior follows the Liénard system, which is widely studied in the field of electrical circuits. The main theorem provides the conditions under which the effect of the impulses causes a stable limit cycle in the Liénard system, even if no limit cycle exists when there are no impulses. The Poincaré–Bendixson theorem for discontinuous dynamical systems and phase plane analysis are used to prove the main theorem. Several examples and their simulations are provided to illustrate the main theorem.

An Existence Result in <i>α</i>-norm for Impulsive Functional Differential Equations with Variable Times

The dynamics of evolving processes is often subjected to abrupt changes such as shocks, harvesting, and natural disasters. Often these short-term perturbations are treated as having acted instantaneously or in the form of “impulses.” In fact, there are many processes and phenomena in the real world, which are subjected during their development to the short-term external influences. Their duration is negligible compared with the total duration of the studied phenomena and processes. Impulsive differential equations take an important place in some area such that physics, chemical technology, population dynamics, biotechnology, and economics. The study of such equations is relatively less developed due to the difficulties created by the state-dependent impulses. In the case of impulses at variable times, a “beating phenomenon” may occur, that is to say, a solution of the differential equation may hit a given barrier several times (including infinitely many times). In this work, we study the existence of solutions for some partial impulsive functional differential equations with variable times in Banach spaces by using the fractional power of closed operators theory. We suppose that the undelayed part admits an analytic semigroup. The delayed part is assumed to be Lipschitz. We use Schaefer fixed-point Theorem to prove the existence of solutions for this first order equation with impulse in <i>α</i>-norm.

Movement of Damped Harmonic Oscillator Affected by State-Dependent Impulses

Movement of Damped Harmonic Oscillator Affected by State-Dependent Impulses

Fixed-time and state-dependent time discontinuities in the theory of Stieltjes differential equations

In the present paper, we are concerned with a very general problem, namely the Stieltjes differential Cauchy problem involving state-dependent discontinuities. Given that the theory of Stieltjes differential equations covers the framework of impulsive problems with fixed-time impulses, in the present work we generalize this setting by allowing the occurrence of fixed-time impulses, as well as the occurrence of state-dependent impulses. Along with an existence result obtained under an overarching set of assumptions involving Stieltjes integrals, it is showed that a least and a greatest solution can be found.

Periodic solutions of superlinear and sublinear state-dependent discontinuous differential equations

A classical, second-order differential equation is considered with state-dependent impulses at both the position and its derivative. This means that the instants of impulsive effects depend on the solutions and they are not fixed beforehand, making the study of this problem more difficult and interesting from the real applications point of view. The existence of periodic solutions follows from a transformation of the problem into a planar system followed by a study of the Poincaré map and the use of some fixed point theorems in the plane. Some examples are presented to illustrate the main results.

The state-dependent impulsive control for a general predator–prey model

In this paper, a general predator–prey model with state-dependent impulse is considered. Based on the geometric analysis and Poincaré map or successor function, we construct three typical types of Bendixson domains to obtain some sufficient conditions for the existence of order-1 periodic solutions. At the same time, the existing domains are discussed with respect to the system parameters. Moreover, the Analogue of Poincaré Criterion is used to obtain the asymptotic stability of the periodic solutions. Finally, to illustrate the results, an example is presented and some numerical simulations are carried out.

Stability Analysis of Stochastic Impulsive Switched Systems with Deterministic State-Dependent Impulses and Switches

This paper studies stability of stochastic impulsive switched systems with deterministic state-dependent impulses and switches. Here, “deterministic state-dependent” means that both switching and i...

Global exponential stability of high-order Hopfield neural networks with state-dependent impulses

In this paper, we discuss the stability of high-order Hopfield neural networks with state-dependent impulses. Under some necessary assumptions, that every solution of the considered system intersects each impulsive surface exactly once is proved. Meanwhile, by using B-equivalence method, the considered system can be simplified to a system with fixed-time impulses. Moreover, some sufficient criteria are derived to ensure the stability between high-order Hopfield neural networks with state-dependent impulses and the corresponding system with fixed-time impulses. The main results show that the stability of high-order Hopfield neural networks with state-dependent impulses maintains no matter the stable continuous subsystems with unstabilizing impulses or the unstable continuous subsystems with stabilizing impulses. Finally, some numerical examples are given to illustrate the effectiveness of our results.

Dynamical Behavior and Bifurcation Analysis of the SIR Model with Continuous Treatment and State-Dependent Impulsive Control

In this study, we propose a state-dependent impulsive model describing the susceptible individuals-triggered interventions. We find that the model with susceptible individuals-guided impulsive interventions can exhibit very complex dynamical behaviors with rich biological meanings. We note that this formulated impulsive model has disease-free periodic solution, and we can investigate the threshold dynamics by defining the control reproduction number. We study the existence and stability of the disease-free periodic solution (DFPS) for [Formula: see text]. Our results show that, even if the basic reproduction number [Formula: see text], the DFPS can still be stable when the threshold level of susceptible population [Formula: see text], indicating that with a proper chosen [Formula: see text], the state-dependent impulsive strategy can effectively control the development of the infectious disease and eradicate the disease eventually. By employing the bifurcation theory, we investigate the bifurcation phenomenon near the DFPS with respect to some key parameters, and observe that a positive order-1 periodic solution can bifurcate from the DFPS via a transcritical bifurcation. By utilizing numerical simulation, we further explore the existence and stability of the positive order-[Formula: see text] periodic solutions, and found the feasibility of stable positive order-1, order-2 and order-3 periodic solutions, that imply the existence of chaos. In particular, we find that there can be three positive order-1 periodic solutions simultaneously, in which one is stable and the other two are unstable. Our finding indicates that the comprehensive strategy combining continuous treatment with state-dependent impulsive vaccination and isolation plays a crucial role in controlling the prevalence and further spread of the infectious diseases.

Existence results for a class of impulsive neutral fractional stochastic integro-differential systems with state dependent delay

In this paper, we provide a framework for study the mild solution to fractional neutral stochastic integro-differential systems with state dependent delay and noninstantaneous impulses in Hilbert spaces. To obtain the existence results, we have utilized the concepts of fractional calculus, fixed point theorems, stochastic analysis and semigroup theory. Finally, examples are provided to demonstrate the obtained results.

Existence and uniqueness of solution for abstract differential equations with state-dependent delayed impulses

We study the existence and uniqueness of mild and classical solutions for a general class of abstract impulsive differential equations with state-dependent impulses. Some examples on partial differential equations are presented.

On Attractors of Generalized Semiflows with Impulses

This paper is devoted to the study of global attractors for problems with state-dependent impulses and possible nonuniqueness of solutions. We provide the criteria under which there exists the global attractor, being on one hand an invariant set, and on the other hand given by the difference of the minimal compact attracting set and the impulsive set M. The new condition (T) used to get the global attractor invariance is discussed and compared with other conditions used in literature for impulsive problems. The theory is illustrated by several examples.