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.     

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.     

On the solvability of implicit differential inclusions

In the paper, the set-valued covering mappings are studied. The statements on solvability, solution estimates, and well-posedness of inclusions with conditionally covering mappings are proved. The results obtained are applied to the investigation of differential inclusions unsolved for the unknown function. The statements on solvability, solution estimates, and well-posedness of these inclusions are derived.     

Chaotic solutions in differential inclusions: chaos in dry friction problems

The existence of a continuum of many chaotic solutions is shown for certain differential inclusions which are small periodic multivalued perturbations of ordinary differential equations possessing homoclinic solutions to hyperbolic fixed points. Applications are given to dry friction problems. Singularly perturbed differential inclusions are investigated as well.

     

Discretizations of Linear Elliptic Partial Differential Inclusions

Differential inclusions provide a suitable framework for modelling choice and uncertainty. In finite dimensions, the theory of ordinary differential inclusions and their numerical approximations is well-developed, whereas little is known for partial differential inclusions, which are the deterministic counterparts of stochastic partial differential equations.

The aim of this article is to analyze strategies for the numerical approximation of the solution set of a linear elliptic partial differential inclusion. The geometry of its solution set is studied, numerical methods are proposed, and error estimates are provided.     

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.     

Approximation and optimization of higher order discrete and differential inclusions

This paper is mainly concerned with the necessary and sufficient conditions of optimality for Cauchy problem of higher order discrete and differential inclusions. Applying optimality conditions of problems with geometric constraints, for arbitrary higher order (say s-order) discrete inclusions optimality conditions are formulated. Also some special transversality conditions, which are peculiar to problems including third order derivatives are formulated. Formulation of sufficient conditions both for convex and non-convex discrete and differential inclusions are based on the apparatus of locally adjoint mappings. Furthermore, an application of these results is demonstrated by solving the problems with third order linear discrete and differential inclusions.     

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.     

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.     

Bartle-Graves Theorem Revisited

We take a fresh look at the Bartle-Graves theorem pointing out the main differences with the standard implicit function theorem. We then present a set-valued version of this theorem which generalizes some recent results. Applications to variational inequalities and differential inclusions are also given.

     

On the existence of solutions for random differential inclusions in a Banach space

We prove two existence theorems for random differential inclusions defined in a separable Banach space. One is about differential inclusions defined on all of the Banach space X and the other for differential inclusion defined on a closed convex subset K. Both theorems are proved through the use of analogous deterministic results, which we also include, and techniques from the theory of measurable multifunctions.     

Nonsmooth analysis on smooth manifolds

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function.

A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

     

Banach空间中微分包含解的存在性*

本文在无穷维Banach空间中讨论微分包含解的存在性,先给出了几个普通微分包含的比较定理,讨论了近似解与解的关系,然后得到了Banach空间中微分包含解的存在性定理.     

Analysis of lumped models with contact and friction

We consider two mathematical models that describe the vibrations of spring-mass-damper systems with contact and friction. In the first model, both the contact and frictional boundary conditions are described with subdifferentials of nonconvex functions. In the second model, the contact is modeled with a Lipschitz continuous function, and the restitution force is described by a differential equation involving a Volterra integral term. The two models lead to second-order differential inclusions with and without an integral term, in which the unknowns are the positions of the masses. For each model, we prove the existence of a solution by using an abstract result for first-order differential inclusions in finite dimensional spaces. For the second model, in addition, we prove the uniqueness of the solution by using a fixed point argument. Finally, we provide examples of systems with contact and friction conditions for which our results are valid.     

Analysis of lumped models with contact and friction

We consider two mathematical models that describe the vibrations of spring-mass-damper systems with contact and friction. In the first model, both the contact and frictional boundary conditions are described with subdifferentials of nonconvex functions. In the second model, the contact is modeled with a Lipschitz continuous function, and the restitution force is described by a differential equation involving a Volterra integral term. The two models lead to second-order differential inclusions with and without an integral term, in which the unknowns are the positions of the masses. For each model, we prove the existence of a solution by using an abstract result for first-order differential inclusions in finite dimensional spaces. For the second model, in addition, we prove the uniqueness of the solution by using a fixed point argument. Finally, we provide examples of systems with contact and friction conditions for which our results are valid.     

On the approximation of solutions of stochastic differential inclusions

A sequence of approximating equations is constructed for stochastic differential inclusions, and the properties of the measures corresponding to solutions of the approximating equations are studied for the class of stochastic differential inclusions.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 43–48, 1986.     

Banach空间中随机微分包含的弱解存在性定理

本文对Banach空间中的微分包含及随机微分包含引入了弱解的概念,并给出了它们的存在性定理。     

On Some Almost-Periodicity Problems in Various Metrics

The Bohr-type and the Bochner-type definitions for almost periodic functions are examined in various metrics (Stepanov, Weyl and Besicovitch). The correct definitions of Besicovitch-like multifunctions are given. Weak almost-periodic solutions are proved for differential equations and inclusions. This problem is also discussed as a fixed-point problem in function spaces.     

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.     

On Systems of Boundary Value Problems for Differential Inclusions

Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.