Brogliato, Bernard Tanwani, Aneel

This survey article addresses the class of continuous-time systems where a system modeled by ordinary differential equations (ODEs) is coupled with a static and time-varying set-valued operator in the feedback. Interconnections of this form model certain classes of nonsmooth systems including sweeping processes, differential inclusions with maximal...

Brogliato, Bernard Goeleven, Daniel

This paper analyzes the existence and uniqueness issues in a class of multivalued Lur'e systems, where the multivalued part is represented as the subdifferential of some convex, proper, lower semicontinuous function. Through suitable transformations the system is recast into the framework of dynamic variational inequalities and the well-posedness (...

Mansour, Mohamed Ait Popovici, Nicolae Théra, Michel
Published in
Positivity

The aim of this paper is to investigate partially ordered real linear topological spaces in which directed sets admit a supremum in their closure. In particular, we point out that this property is intimately related to the normality of the ordering cone and also to the Scott continuity of functionals belonging to the nonnegative polar of the orderi...

Outrata, J. V. Römisch, W.
Published in
Journal of Optimization Theory and Applications

The paper deals with the minimization of an integral functional over an Lp space subject to various types of constraints. For such optimization problems, new necessary optimality conditions are derived, based on several concepts of nonsmooth analysis. In particular, we employ the generalized differential calculus of Mordukhovich and the fuzzy calcu...

Herings, P. J. J. Koshevoy, G. A. Talman, A. J. J. Yang, Z.
Published in
Journal of Optimization Theory and Applications

Let X be a nonempty, compact, convex set in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}^n$$ \end{document} and let φ be an upper semicontinuous mapping ...

Ward, D.E. Lee, G.M.
Published in
Annals of Operations Research

We obtain an upper bound for the upper subderivative of the marginal function of an abstract parametric optimization problem when the objective function is lower semicontinuous. Moreover, we apply the result to a nonlinear program with right-hand side perturbations. As a result, we obtain an upper bound for the upper subderivative of the marginal f...

Colombo, Giovanni Goncharov, Vladimir V.
Published in
Set-Valued Analysis

We study the sweeping processes in a Hilbert space which are generated by a closed not necessarily convex moving set. A technique is developed, based on measurability properties of normal cones, in order to prove existence of solutions. Some existence results are proved, with or without hypothesis of compactness; moreover, under suitable assumption...

Fabian, Marián Mordukhovich, Boris S.
Published in
Set-Valued Analysis

We show that the Asplund property of Banach spaces is not only sufficient but also a necessary condition for the fulfillment of some basic results in nonsmooth analysis involving Fréchet-like normals and subdifferentials as well as their sequential limits. In this way we obtain new characterizations of Asplund spaces within the framework of nonsmoo...

Rockafellar, Ralph

Let $C$ be a nonempty closed subset of $\mathbb{R}^n$. For each $x \in C$, the tangent cone $T_C(x)$ in the sense ofClarke consists of all $y \in \mathbb{R}^n$ such that, whenever one has sequences $t_k\downarrow 0$ and $x_k \rightarrow x$ with $x_k \in C$, there exist $y_k \rightarrow y$ with $x_k + t_ky_k \in C$ for all $k$. This is not Clarke’s ...