Periodic solutions of nonlinear systems of differential-operator equations with impulse action

Numerical-analytic methods are considered for the investigation of the existence and the approximate construction of periodic solutions of nonlinear differential-operator equations, subjected to an impulse action.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1260–1264, September, 1991.     

Asymptotic stability and instability of the solutions of systems with impulse action

In this paper, we study the stability of the zero solution of a system of ordinary differential equations subject to impulse action. Using the method of Lyapunov functions, we obtain tests for asymptotic stability or instability of the system. Illustrative examples are given.     

Linear Noetherian boundary value problems for differential systems with an impulse action

A solvability criterion of linear nonhomogeneous boundary value problems is obtained for systems of ordinary differential equations with impulse action in the general case when the number of boundary conditions does not coincide with the order of the differential system (Noetherian problems). The generalized Green operator of such boundary value problems is constructed. Its connection with the generalized inverse operator of the operator of the original boundary value problem is shown.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 564–568, April, 1992.     

Stability of solutions of pseudolinear differential equations with impulse action

In this paper, we develop a new approach to constructing piecewise differentiable Lyapunov functions for certain classes of nonlinear differential equations with impulse action. This approach is based on the method of “frozen” coefficients, and the required function is constructed as a pseudoquadratic form. For the case under consideration, stability conditions in the sense of Lyapunov are obtained. The proposed approach can be used to study the stability of the critical equilibrium states of systems of differential equations with impulse action.     

Averaging of impulse evolution systems

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 76–83, January, 1992.     

Periodic solutions of weakly nonlinear systems with impulse interaction

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 622–626, May, 1989.     

Numerical study of mathematical models of micromechanics with periodic impulse action

We consider the results of the numerical study of the mathematical models of two microelectromechanical systems (MEMS). The models as initial boundary value problems describe the cylindrical flexure of an elastic beam as a movable electrode under action of a repetitive intensity impulse of the electrostatic field between the movable and fixed electrodes in a microgap. In the first problem, both ends of the beamare rigidly fixed, while in the second problemwe consider a cantilever beam. The range of the parameters is found where the model has two periodic solutions with periods of impulse action one of which is stable and the other is unstable.     

Stability with respect to part of the variables in systems with impulse effect

Effective sufficient conditions are found for stability with respect to part of the variables in systems of ordinary differential equations with impulse effect. The approach presented is based on the specially introduced piecewise continuous Ljapunov functions.     

Integro-sum inequalities and motion stability of systems with impulse perturbations

This article is devoted to the investigation of the properties of motion represented by essential nonlinear system of differential equations with impulse effect. The results, based on the solvability problem by S.A. Chaplygin for discontinuous functions, satisfy “integro-sum” inequalities. The conditions of boundedness, stability, asymptotical stability, practical stability, attraction motion for different kinds of nonlinearities of systems are obtained.     

Input-to-state stability of impulsive systems with hybrid delayed impulse effects

The goal of this paper is to study properties of input-to-state stability (ISS) and integral input-to-state stability (iISS) of impulsive systems with hybrid delayed impulses, and a set of Lyapunov-based sufficient conditions ensuring ISS/iISS properties are obtained. Those conditions reveal the effects of hybrid delayed impulses on ISS/iISS and establish the relationship between impulsive frequency and the time delay existing in hybrid impulses. When the continuous dynamics of the system are stabilizing, the ISS property can be retained under the impulse scheme even if there exist destabilizing impulses. Conversely, when the impulse dynamics are stabilizing, but the continuous dynamics are not, the ISS property can be obtained if the interval between impulses are not overly long. Two illustrative examples are presented, with their numerical simulations, to demonstrate the effectiveness of the main results.     

Consensus seeking in multi-agent systems via hybrid protocols with impulse delays

This paper studies the consensus problem of multi-agent systems with both fixed and switching topologies. A hybrid consensus protocol is proposed to take into consideration of continuous-time communications among agents and delayed instant information exchanges on a sequence of discrete times. Based on the proposed algorithms, the multi-agent systems with the hybrid consensus protocols are described in the form of impulsive systems or impulsive switching systems. By employing results from matrix theory and algebraic graph theory, some sufficient conditions for the consensus of multi-agent systems with fixed and switching topologies are established, respectively. Our results show that, for small impulse delays, the hybrid consensus protocols can solve the consensus problem if the union of continuous-time and impulsive-time interaction digraphs contains a spanning tree frequently enough. Simulations are provided to demonstrate the effectiveness of the proposed consensus protocols.     

On the stability of invariant sets of systems with impulse effect

A system of differential equations with impulse effect is considered. It is assumed that this system has an invariant set M$M$. By means of the direct Lyapunov method, the necessary and sufficient conditions of its uniform asymptotic stability are obtained. The conditions on the perturbations of right hand sides of differential equations and impulse effects, under which the uniform asymptotic stability of the invariant set M$M$ of the “nonperturbed” system implies the uniform asymptotic stability of the invariant set of the “perturbed” system, are obtained. The stability properties of invariant sets of periodic systems are also studied.     

Exponential dichotomy of solutions of impulse systems

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 6, pp. 779–783, June, 1989.     

Periodic solutions of singularly-perturbed systems of differential equations with impulse effect

Summary The present paper considers the problem for existence, uniqueness and asymptotic representation of periodic solutions of some singularly-perturbed systems of differential equations with impulses. The study is carried out with the help of a modification of the boundary functions method, adequately worked out to match the specific issues of the considered problem.

---

Zusammenfassung Wir betrachten das Problem der Existenz, Eindeutigkeit und der asymptotischen Darstellung periodischer Lösungen singulär gestörter Systeme von Differentialgleichungen mit Impulsen. Die Studie wird mittels speziell dem Problem angepaßter Modifikation der Randwertfunktionsmethode ausgeführt.