Search

It is shown that by knowing the slope of a bounded from below lower semicontinuous convex function at each point of a Banach space, we know the function up to an additive constant. [ABSTRACT FROM AUTHOR]

In this paper, we analyze and discuss the well-posedness of a new variant of the so-called sweeping process. In this variant, the prox-regular moving set C(t) is supposed to have an absolutely continuous variation and is perturbed by an integral forcing term. The integrand of the forcing term depends on two time-variables; that is, we study a general integro-differential sweeping process of Volterra type. The results are applied to nonregular electrical circuits containing timevarying capacitors and nonsmooth electronic devices like diodes. A circuit with transmission line, diode, and inductor is also analyzed. All these applications represent an additional novelty of our paper. [ABSTRACT FROM AUTHOR]

This paper is devoted to nonconvex/prox-regular separations of sets in Hilbert spaces. We introduce the Legendre-Fenchel r-conjugate of a prescribed function and r-quadratic support functionals and points of a given set, all associated to a positive constant r. By means of these concepts we obtain nonlinear functional separations for points and prox-regular sets. In addition to such functional separations, we also establish geometric separation results with balls for a prox-regular set and a strongly convex set. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence and uniqueness of solutions for perturbed sweeping process measure differential inclusions in infinite dimensional setting. The possibly unbounded moving set is prox-regular and controlled only through the truncated Hausdorff-Pompeiu distance. The normal cone involved is perturbed by a kind of Carathéodory mapping satisfying a time-dependent hypomonotonicity assumption on bounded sets. Various properties of the solution mapping are also provided. [ABSTRACT FROM AUTHOR]

Projected differential equations are known as fundamental mathematical models in economics, for electric circuits, etc. The present paper studies the (higher order) derivability as well as a generalized type of derivability of solutions of such equations when the set involved for projections is prox-regular with smooth boundary. [ABSTRACT FROM AUTHOR]

We consider a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real-valued target function defined on a Banach space. If the target function is the indicator function of a closed set, this problem is a minimal time problem for a target set, studied previously in particular by Colombo, Goncharov and Mordukhovich. We investigate several properties of the Fréchet and proximal subdifferentials for the infimum time function. Also explicit expressions of the above mentioned subdifferentials as well as various directional derivatives are obtained. We provide some examples to show the essentiality of assumptions of our theorems. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate the prox-regularity behaviour of solution mappings to generalized equations. This study is realized through a nonconvex uniform Robinson−Ursescu type theorem. Then, we derive new significant results for the preservation of prox-regularity under various and usual set operations. The role and applications of prox-regularity of solution sets of generalized equations are illustrated with dynamical systems with constraints. [ABSTRACT FROM AUTHOR]

We continue the investigation of a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real-valued target function defined on a Banach space. In this paper we obtain an explicit description of the minimal time projection set in terms of the Legendre--Fenchel conjugate function and provide some sufficient conditions for this set to be nonempty. The single valuedness and Lipschitz property of the minimal time projection are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness (in the sense of existence and uniqueness of a solution) of a discontinuous sweeping process involving prox-regular sets in Hilbert spaces. The variation of the moving set is controlled by a positive Radon measure and the perturbation is assumed to satisfy a Lipschitz property. The existence of a solution with bounded variation is achieved thanks to the Moreau's catching-up algorithm adapted to this kind of problem. Various properties and estimates of jumps of the solution are also provided. We give sufficient conditions to ensure the uniform proxregularity when the moving set is described by inequality constraints. As an application, we consider a nonlinear differential complementarity system which is a combination of an ordinary differential equation with a nonlinear complementarily condition. Such problems appear in many areas such as nonsmooth mechanics, nonregular electrical circuits and control systems. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a perturbed differential inclusion governed by a sweeping process in a Hilbert space. The sweeping process is perturbed by a sum of a single-valued map satisfying a Lipschitz condition and a scalarly upper semicontinuous set-valued map. [ABSTRACT FROM AUTHOR]

In this paper, we derive exact chain rules for a proper epsilon-subdifferential in the sense of Benson of extended vector mappings, recently introduced by ourselves. For this aim, we use a new regularity condition and a new strong epsilon-subdifferential for vector mappings. In particular, we determine chain rules when one of the mappings is linear, obtaining formulations easier to handle in the finite-dimensional case by considering the componentwise order. This Benson proper epsilon-subdifferential generalizes and improves several of the most important proper epsilon-subdifferentials of vector mappings given in the literature and, consequently, the results presented in this work extend known chain rules stated for the last ones. As an application, we derive a characterization of approximate Benson proper solutions of implicitly constrained convex Pareto problems. Moreover, we estimate the distance between the objective values of these approximate proper solutions and the set of nondominated attained values. [ABSTRACT FROM AUTHOR]

Abstract: In a vector space endowed with a uniformly Gâteaux differentiable norm, it is proved that the Moreau envelope enjoys many remarkable differential properties and that its subdifferential can be completely described through a certain approximate proximal mapping. This description shows in particular that the Moreau envelope is essentially directionally smooth. New differential properties are derived for the distance function associated with a closed set. Moreover, the analysis, when applied to the investigation of the convexity of Tchebyshev sets, allows us to recover several known results in the literature and to provide some new ones. [Copyright &y& Elsevier]

For a locally Lipschitz integral functional If on L¹(T, Rn associated with a measurable integrand f, the limiting subdifferential and the approximate subdifferential never coincide at a point x0where f(t, ⋅) is not subdifferentially regular at x0 (t) for a.e. t Ε T. The coincidence of both subdifferentials occurs on a dense set of L¹(T, Rn) if and only if f(t, ⋅) is convex for a.e. t Ε T. Our results allow us to characterize Aubin's Lipschitz-like property as well as the convexity of multivalued mappings between L¹-spaces. New necessary optimality conditions for some Bolza problems are also obtained. [ABSTRACT FROM AUTHOR]

In a separable Banach space we consider a three point boundary value problem for a second order differential inclusion of the form [image omitted] We investigate the existence of solutions, when the multifunction F is unbounded-valued and satisfies a pseudo-Lipschitz property. Then, a Lipschitz case is derived and the associated relaxed problem is studied. [ABSTRACT FROM AUTHOR]