Local controllability for autonomous nonlinear systems

This paper is concerned with the study of local controllability for nonlinear, analytic, autonomous, control systems. To this aim, we consider the control system as a set-valued differential system, and we show that the convexification of the set of velocities does not change the local controllability properties. Given a finite family of vector fields, we also give a necessary condition for local controllability at timet, for everyt.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities

The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.     

Characterizations of Solutions for Vector Equilibrium Problems

In this paper, we characterize the solutions of vector equilibrium problems as well as dual vector equilibrium problems. We establish also vector optimization problem formulations of set-valued maps for vector equilibrium problems and dual vector equilibrium problems, which include vector variational inequality problems and vector complementarity problems. The set-valued maps involved in our formulations depend on the data of the vector equilibrium problems, but not on their solution sets. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of the vector equilibrium problems.     

集值Markov链及其随机集落影性质

提出了一种集值Markov链,该模型是对基于随机变量的Markov链推广,将随机变量提升到随机集上.模型继承了经典Markov链的诸多良好性质,而且可以退化为经典的Markov链模型.为了进一步分析该模型,引入随机集落影理论,提出转移落影、落影分布等概念,并给出了部分结论和性质.最后,给出一个应用实例.     

Levitin-Polyak Well-Posedness of Constrained Inverse Quasivariational Inequality

In this article, we introduce and study different types of Levitin–Polyak well-posedness for a constrained inverse quasivariational inequality problem. Criteria and characterizations for these types of well-posedness for inverse quasivariational inequality problems are given. Su?cient conditions for the Levitin–Polyak well-posedness of inverse quasivariational inequality problems are also established.     

Some useful set-valued maps in set optimization

In this article, we introduce several classes of set-valued maps which can be useful in set optimization due to their applications. Exactly, we present some set-valued maps defined by scalar and vector functions and study their properties such as continuity and convexity among others. In addition, we compute their asymptotic maps which can be employed to establish coercivity and existence results in the framework of set optimization problems. Finally, we expose some possible directions for further research.     

Optimization of higher order differential inclusions with initial value problem

This paper concerns the sufficient conditions of optimality for initial value problem with higher order differential inclusions (HODIs) and free endpoint constraints. Formulation of the transversality conditions plays a substantial role in the next investigations without which hardly any necessary or sufficient conditions would be obtained. In terms of Euler–Lagrange and Hamiltonian forms the sufficient conditions of optimality both for convex and “non-convex” HODIs are based on the apparatus of locally adjoint mappings. Moreover, by applying the main result to a Bolza problem described by a polynomial differential operator with constant coefficients in terms of the adjoint differential operator the sufficient condition of optimality is obtained.     

A revised pre-order principle and set-valued Ekeland variational principles with generalized distances

In my former paper "A pre-order principle and set-valued Ekeland variational principle"(see [J. Math. Anal. Appl., 419, 904–937(2014)]), we established a general pre-order principle.From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles(denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy's EVP in [On generalized Ekeland's variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381–396(2011)], where the perturbation contains a weak τ-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle,but also can be applied to obtain the above Khanh and Quy's EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.     

New constructions for local approximation of Lipschitz functions.II

In this paper, some properties of the set-valued mapping _Dαf(.)$D_{\alpha }f(.)$ connected with the new approximation method of a function f(.)$f(.)$ defined in the first part of the article are given. Continuity and Lipschitz properties of _Dαf(.)$D_{\alpha }f(.)$ are formulated. A continuous extension of the Clarke subdifferential of any function represented as a difference of two convex functions is given. For the convex case, the set-valued mapping _Dαf(.)$D_{\alpha }f(.)$ is similar to the ε$\epsilon$-subdifferential mapping.