共查询到20条相似文献,搜索用时 31 毫秒
1.
The averaging method is one of the most powerful methods used to analyse differential equations appearing in the study of nonlinear problems. The idea behind the averaging method is to replace the original equation by an averaged equation with simple structure and close solutions. A large number of practical problems lead to differential equations with discontinuous right-hand sides. In a rigorous theory of such systems, developed by Filippov, solutions of a differential equation with discontinuous right-hand side are regarded as being solutions to a special differential inclusion with upper semi-continuous right-hand side. The averaging method was studied for such inclusions by many authors using different and rather restrictive conditions on the regularity of the averaged inclusion. In this paper we prove natural extensions of Bogolyubov’s first theorem and the Samoilenko-Stanzhitskii theorem to differential inclusions with an upper semi-continuous right-hand side. We prove that the solution set of the original differential inclusion is contained in a neighbourhood of the solution set of the averaged one. The extension of Bogolyubov’s theorem concerns finite time intervals, while the extension of the Samoilenko-Stanzhitskii theorem deals with solutions defined on the infinite interval. The averaged inclusion is defined as a special upper limit and no additional condition on its regularity is required. 相似文献
2.
In this work we propose a method to study a weak exponential stability for time-varying differential inclusions applying an averaging procedure to a first approximation. Namely, we show that a weak exponential stability of the averaged first approximation to the differential inclusion implies the weak exponential stability of the original time-varying inclusion. The result is illustrated by an example. 相似文献
3.
We consider the problem of application of the averaging method to the asymptotic approximation of solutions of differential inclusions of standard form in the case where the average of the right-hand side does not exist. 相似文献
4.
Götz Grammel 《Set-Valued Analysis》1996,4(4):361-374
We consider nonlinear, singularly perturbed differential inclusions and apply the averaging method in order to construct a limit differential inclusion for slow motion. The main approximation result states that the existence and regularity of the limit differential inclusion suffice to describe the limit behavior of the slow motion. We give explicit approximation rates for the uniform convergence on compact time intervals. The approach works under controllability or stability properties of fast motion. 相似文献
5.
We justify a method of complete and partial averaging on finite and infinite intervals for differential inclusions with many-valued pulses.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1526–1532, November, 1995. 相似文献
6.
Ido Bright 《Set-Valued and Variational Analysis》2014,22(4):843-857
We present new tight bounds for averaging differential inclusions, which we apply to multi-frequency inclusions consisting of a sum of time periodic set-valued mappings. For this family of inclusions we establish a tight estimate of order O (??) on the approximation error. These results are then applied to control systems consisting of a sum of time-periodic functions. 相似文献
7.
We justify partial averaging schemes for quasidifferential equations that generalize the corresponding results for differential equations and inclusions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1442–1447, October, 1995. 相似文献
8.
Junfeng Zhang Zhengzhi Han Jun Huang 《Communications in Nonlinear Science & Numerical Simulation》2013,18(10):2790-2800
This paper is concerned with the stabilization of differential inclusions. By using control Lyapunov functions, a design method of homogeneous controllers for differential equation systems is first addressed. Then, the design method is developed to two classes of differential inclusions without uncertainties: homogeneous differential inclusions and nonhomogeneous ones. By means of homogeneous domination theory, a homogeneous controller for differential inclusions with uncertainties is constructed. Comparing to the existing results in the literature, an improved formula of homogeneous controllers is proposed, and the difficulty of the controller design for uncertain differential inclusions is reduced. Finally, two simulation examples are given to verify the preset design. 相似文献
9.
Averaging schemes for functional differential inclusions in Banach spaces with slow and fast time variables are studied. Under mild suppositions on the regularity, the periodic case and the case of non-existence of an average are investigated. The accuracy of the averaging technique is considered as well. In particular, for periodic systems, the usual linear approximation is achieved. Under stronger regularity conditions, approximation orders for Krylov-Bogoliubov-Mitropolskii type right-hand sides are derived. 相似文献
10.
A version of the statistiad method of averaging the system of equilibrium equations for an elastoplastic two-component composite material in order to predict its macroscopic non-linear hardening is proposed. This version, unlike the averaging method developed previously in [l, 2], enables one to model and estimate the degree of connectedness of the matrix and the inclusions and to take into account the non-uniformity of the distribution and the development of plastic deformations. Macroscopic governing equations are constructed which describe the non-linear hardening of a composite material outside the elasticity limit, and its effective characteristics are calculated. 相似文献
11.
I. A. Finogenko 《Siberian Mathematical Journal》2014,55(2):372-386
Considering nonautonomous differential inclusions we introduce the concept of limit differential inclusions, study their properties and invariance-type properties of the ω-limit sets of solutions, and establish an analog of La Salle’s invariance principle using Lyapunov functions with the derivatives of constant sign. The method is equally applicable to differential equations and, under appropriate assumptions, yields some previously-available results. 相似文献
12.
Elimhan N. Mahmudov 《Journal of Optimization Theory and Applications》2018,177(2):345-375
The present paper studies a new class of problems of optimal control theory with Sturm–Liouville-type differential inclusions involving second-order linear self-adjoint differential operators. Our main goal is to derive the optimality conditions of Mayer problem for differential inclusions with initial point constraints. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximation inclusions are investigated. Necessary and sufficient conditions, containing both the Euler–Lagrange and Hamiltonian-type inclusions and “transversality” conditions are derived. The idea for obtaining optimality conditions of Mayer problem is based on applying locally adjoint mappings. This approach provides several important equivalence results concerning locally adjoint mappings to Sturm–Liouville-type set-valued mappings. The result strengthens and generalizes to the problem with a second-order non-self-adjoint differential operator; a suitable choice of coefficients then transforms this operator to the desired Sturm–Liouville-type problem. In particular, if a positive-valued, scalar function specific to Sturm–Liouville differential inclusions is identically equal to one, we have immediately the optimality conditions for the second-order discrete and differential inclusions. Furthermore, practical applications of these results are demonstrated by optimization of some “linear” optimal control problems for which the Weierstrass–Pontryagin maximum condition is obtained. 相似文献
13.
Systems of difference equations containing small parameters are studied by a constructive perturbation scheme analogous to the one developed by the authors for the study of differential equations. The method results in an averaging procedure for difference equations, and it is particularly well suited to certain highly oscillatory, nonlinear systems. The method is applied to problems from population genetics, pattern recognition, and the numerical analysis of stiff differential equations 相似文献
14.
The effects of orientation and shape of filler particles on the elastic properties of composites have been analyzed. The elastic
constants of a composite with irregularly oriented filler particles were calculated by using the method of orientational averaging
of the properties of a representative structural element. The elastic constants of the structural element were found according
to a known generalized Eshelby solution for a finite concentration of ellipsoidal inclusions. The diagrams of elasticity anisotropy
for a transversely isotropic structural element and an orthotropic composite with irregularly oriented inclusions are presented.
A quantitative estimate for the degree of anisotropy of elastic properties of composites is suggested. Data on the influence
of shape anisometry of inclusions on the anisotropy coefficient of filled composites are also reported. 相似文献
15.
Sascha Trostorff 《Mathematical Methods in the Applied Sciences》2015,38(5):834-850
We study integro‐differential inclusions in Hilbert spaces with operator‐valued kernels and give sufficient conditions for the well‐posedness. We show that several types of integro‐differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well‐posedness within this framework. As an example, we apply our results to the equations of visco‐elasticity and to a class of nonlinear integro‐differential inclusions describing phase transition phenomena in materials with memory. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
16.
17.
An existence theorem for differential inclusions in Hilbert spaces with nonlocal conditions is proved. Periodic, anti-periodic, mean value and multipoint conditions are included in this study. The investigation is based on a combination of the approximation solvability method with Hartman-type inequalities. A feedback control problem associated to a first order partial differential equation completes this discussion. 相似文献
18.
We study differential inclusions with existence which are natural approximations of differential inclusions without existence. In a Banach space setting, we also establish a multivalued version of a recent result of Colombo and Garay [3]. 相似文献
19.
Implicit function theorems are derived for nonlinear set valued equations that satisfy a relaxed one-sided Lipschitz condition.
We discuss a local and a global version and study in detail the continuity properties of the implicit set-valued function.
Applications are provided to the Crank–Nicolson scheme for differential inclusions and to the analysis of differential algebraic
inclusions. 相似文献
20.
Ido Bright 《Journal of Differential Equations》2009,246(7):2922-1284
Estimates are presented for the averaging method of ordinary differential equations. Previous results are improved by relaxing the conditions under which they hold, and by providing tight bounds for the estimations for almost-periodic differential equations and quasi-periodic differential equations. 相似文献