On Selections of Set-Valued Inclusions in a Single Variable with Applications to Several Variables

We present some applications of the result corresponding to the existence of a unique selection of a set-valued function satisfying inclusions in a single variable to the inclusions in several variables, especially the general linear inclusions or quadratic inclusions.     

Homogeneous feedback design of differential inclusions based on control Lyapunov functions

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.     

Set-valued quasi variational inclusions

In this paper, we introduce and study a new class of variational inclusions, called the set-valued quasi variational inclusions. The resolvent operator technique is used to establish the equivalence between the set-valued quasi variational inclusions and the fixed point problem. This equivalence is used to study the existence of a solution and to suggest a number of iterative algorithms for solving the set-valued variational inclusions. We also study the convergence criteria of these algorithms.     

Stochastic Equations and Inclusions with Mean Derivatives and Some Applications

The paper is devoted to a brief introduction into the theory of equations and inclusions with mean derivatives and to investigation of a special type of such inclusions called inclusions of geometric Brownian motion type. The existence of optimal solutions maximizing some cost criteria, is proved.     

Problems of Thin Inclusions in a Two-Dimensional Viscoelastic Body

Under study are the equilibrium problems for a two-dimensional viscoelastic body with delaminated thin inclusions in the cases of elastic and rigid inclusions. Both variational and differential formulations of the problems with nonlinear boundary conditions are presented; their unique solvability is substantiated. For the case of a thin elastic inclusion modelled as a Bernoulli–Euler beam, we consider the passage to the limit as the rigidity parameter of the inclusion tends to infinity. In the limit it is the problem about a thin rigid inclusion. Relationship is established between the problems about thin rigid inclusions and the previously considered problems about volume rigid inclusions. The corresponding passage to the limit is justified in the case of inclusions without delamination.     

Homogenization of Unidirectional and Arbitrarily Oriented Fiber-Reinforced Materials by the Method of Conditional Moments

In this paper the method of conditional moments is developed for the case of a two–component matrix composite with randomly distributed unidirectional and arbitrarily oriented ellipsoidal inclusions. The algorithm for determination of the effective elastic properties of composites from the given elastic constants of the components and geometrical parameters and orientation of inclusions is discussed. It is assumed that the components of the composite show orthotropic symmetry of thermoelastic properties. As a numerical example arbolite (straw particle inclusions in a cement matrix) is considered. The dependencies of Young's moduli, Poisson's ratios and shear moduli from the concentration of inclusions and for certain orientations of the inclusions are predicted and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

微分包含的周期生存轨道

对微分包含的周期生存轨道进行了研究讨论。首先给出微分包含生存问题的一约化性质;然后,利用投影微分包含的方法给出有限维空间中微分包含的周期生存轨道的一个存在性结果;在此基础上,利用Galerkin逼近方法得到Hilbert空间中偏微分包含周期生存轨道的存在性定理。     

Subdifferential analysis of differential inclusions via discretization

The framework of differential inclusions encompasses modern optimal control and the calculus of variations. Necessary optimality conditions in the literature identify potentially optimal paths, but do not show how to perturb paths to optimality. We first look at the corresponding discretized inclusions, estimating the subdifferential dependence of the optimal value in terms of the endpoints of the feasible paths. Our approach is to first estimate the coderivative of the reachable map. The discretized (nonsmooth) Euler–Lagrange and Transversality Conditions follow as a corollary. We obtain corresponding results for differential inclusions by passing discretized inclusions to the limit.     

On integro‐differential inclusions with operator‐valued kernels

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.     

Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions

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.     

Tight Bounds for Averaging Multi-Frequency Differential Inclusions,Applied to Control Systems

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.     

Asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities

We consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. In this paper we derive an asymptotic formula of the solution, which characterizes the gradient blow-up of the solution in terms of conductivities of inclusions as well as the distance between inclusions. The asymptotic formula is expressed in bipolar coordinates in terms of the Lerch transcendent function, and it is valid for inclusions with arbitrary constant conductivities. We illustrate our results with numerical calculations.     

Transversality condition and optimization of higher order ordinary differential inclusions

In the present paper, a Bolza problem of optimal control theory with a fixed time interval given by convex and nonconvex second-order differential inclusions (P_H) is studied. Our main goal is to derive sufficient optimality conditions for Cauchy problem of sth-order differential inclusions. The sufficient conditions including distinctive transversality condition are proved incorporating the Euler–Lagrange and Hamiltonian type inclusions. The basic concepts involved in obtaining optimality conditions are the locally adjoint mappings. Furthermore, the application of these results is demonstrated by solving the problems with third-order differential inclusions.     

Dual boundary element method for problems of the theory of thin inclusions

We have proposed to apply the dual boundary element method in problems of the theory of thin inclusions. The contact conditions on the boundary of a thin inclusion are considered as jumps of displacements and stresses in the body on the median surface of this defect. Thus, the relations between the unknown discontinuities and average values of the displacements and stresses are a model of inclusions. For rectilinear boundary elements, we have constructed models of inclusions, taking into account the tension, shear, and bending of a thin inclusion. Examples for rectilinear and curved inclusions have been considered. Comparison of the results obtained by the proposed technique with data based on the direct approach shows the efficiency of the proposed method.     

The glueing construction and double categories

We introduce Artin–Wraith glueing and locally closed inclusions in double categories. Examples include locales, toposes, topological spaces, categories, and posets. With appropriate assumptions, we show that locally closed inclusions are exponentiable, and the exponentials are constructed via Artin–Wraith glueing. Thus, we obtain a single theorem establishing the exponentiability of locally closed inclusions in these five cases.     

On certain classes of functional inclusions with causal operators in Banach spaces

We introduce the notion of a multivalued causal operator and consider an abstract Cauchy problem in a Banach space for various classes of functional inclusions with causal operators. The methods of the topological degree theory for condensing maps are applied to obtain local and global existence results for this problem and to study the continuous dependence of a solution set on initial data. As application we generalize some existence results for semilinear functional differential inclusions and Volterra integro-differential inclusions with delay.     

Method of Riemann surfaces for an inverse antiplane problem in an n-connected domain

ABSTRACT

An inverse problem of the theory of harmonic functions for an n-connected domain is analyzed. The problem is equivalent to a problem of antiplane elasticity on determination of the profiles of n uniformly stressed inclusions. The inclusions are in ideal contact with the surrounding matrix, the stress field inside the inclusions is uniform, and at infinity the body is subjected to antiplane uniform shear. The exterior of the inclusions, an n-connected domain, is treated as the image by a conformal map of an n-connected slit domain with the slits lying in the same line. The inverse problem is solved by quadratures by reducing it to two Riemann-Hilbert problems on a Riemann surface of genus n?1. Samples of two and three symmetric and non-symmetric uniformly stressed inclusions are reported.     

Formal asymptotics of eigenmodes for oscillating elastic spatial bodies with concentrated masses

Limit spectral problems are derived for the problem on oscillations of a solid with small heavy (or light) inclusions. The asymptotic ansatzs for eigenvalues and eigenvectors, as well as the limit problems, are crucially dependent on both the relation between the geometric and physical parameters and the disposition of the inclusions. It is established that, for heavy inclusions, the limit problems are united into a more complex resultant problem describing the “far action” in the set of inclusions. Bibliography: 39 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 31–76.     

Emarks on Differential Inclusions Without Existence or Continuous Dependence

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].     

Multivalent guiding function in a problem on existence of periodic solutions of some classes of differential inclusions

We propose to use the multivalent guiding function for the study of periodic solutions of some classes of differential inclusions. More precisely, we consider the periodic problem for nonlinear systems described by differential inclusions with both convex and nonconvex right-hand side. The latter include differential inclusions with a regular right-hand side. Note that the class of regular multimaps is wide enough. It includes, for example, bounded almost lower semicontinuous multimaps with compact values.