Exponential dichotomy of solutions of impulse systems

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

Periodic solutions of quasilinear impulse systems in the critical case

Conditions for the existence of periodic solutions of weakly-nonlinear autonomous and nonautonomous impulse systems in the critical case are determined.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 308–315, March, 1991.     

Periodic solutions of quasilinear impulse systems in the critical case

Conditions for the existence of periodic solutions of weakly-nonlinear autonomous and nonautonomous impulse systems in the critical case are determined.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 308–315, March, 1991.     

Representation of solutions of control hybrid differential-difference impulse systems

We consider linear nonstationary hybrid differential-difference controlled dynamical systems under the action of impulses. For such systems, we derive integral representations of solutions on the basis of the solution of dual (adjoint) systems. This is, in a sense, a generalization to these systems of the Cauchy formula for ordinary systems. This formula is refined in the stationary case. The results are illustrated by an example     

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.     

Generalized solutions to nonlinear first-order systems

In the present work we reinterpret a result of approximate solutions to a nonlinear first order system in the framework of Colombeau's theory, defining an algebra of generalized germs where the problem $$u_t = f(x,t,u,u_x ), u|_{t = 0} = u_o (x)$$ has a unique solution, wheref andu ₀ are vector-valued functions.     

Generalized solutions to semilinear hyperbolic systems

In this article semilinear hyperbolic first order systems in two variables are considered, whose nonlinearity satisfies a global Lipschitz condition. It is shown that these systems admit unique global solutions in the Colombeau algebraG(²). In particular, this provides unique generalized solutions for arbitrary distributions as initial data. The solution inG(²) is shown to be consistent with the locally integrable or the distributional solutions, when they exist.     

Periodic solutions of weakly nonlinear systems with impulse interaction

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

Continuous relationship between a parameter and the solutions of impulse evolution systems

A theorem on continuous dependence of the solutions of a nonlinear impulse evolution system on a parameter is proved; this theorem can be used as a basis for an averaging principle.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1078–1083, August, 1992.     

Differentiable dependence of the solutions of impulse systems on initial data

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 8, pp. 1028–1033, August, 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.

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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.

     

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.     

Generalized solutions in systems with active unilateral constraints

The concept of the generalized solution that admits well-posed representation of controlled complex behavior in systems with active unilateral phase constraints is proposed. Based on this concept, the definition of the generalized solution for this class of problems is introduced that encompasses Zeno type behavior and sliding modes along the constraint boundary. The general representation of such solutions in terms of nonlinear differential equations with a measure is derived. The latter is shown to solve a long-standing problem of providing unique extensibility of a trajectory beyond accumulation points in systems with Zeno-type behavior. An example is given, showing that the representation proposed completely captures Zeno-type behavior and provides unique extensibility of solutions without the need to truncate infinite sequences and/or switch system coefficients depending on system motion relative to the generalized coordinates of the accumulation point.     

Generalized almost periodic solutions and ergodic properties of quasi-autonomous dissipative systems

Let H be a real Hilbert space, A a maximal monotone operator in H and $\text{? R → H}$ a measurable function which is S¹-almost periodic. Assuming that the positive trajectories of the equation $\text{du}\text{dt}\text{+ Au(t)? f(t)}$ are bounded (in H) for t ? 0, we construct a kind of generalized almost periodic “solution” of the equation, and we show how to deduce information of ergodic type on the asymptotic behavior of the trajectories as t → +∞.     

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.     

Existence of periodic solutions of nonlinear systems of differential equations with impulse effect

Sufficient conditions for the existence of periodic solutions of nonlinear systems of differential equations with impulses in a canonic domain have been found in the paper.     

Lyapunov's second method in problems of the stability of solutions of systems with impulse effect

A system of ordinary differential equations with impulse effect at fixed moments of time is considered. The system is assumed to have the zero solution. It is shown that the existence of a corresponding Lyapunov function is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution. Restrictions on perturbations of the right-hand sides of differential equations and impulse effects are obtained under which the uniform asymptotic stability of the zero solution of the ‘unperturbed’ system implies the uniform asymptotic stability of the zero solution of the ‘perturbed’ system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)