the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

clarke coderivatives of efficient point multifunctions in parametric vector optimization abstract the paper is devoted to the study of the clarke circatangent coderivatives of the efficient point multifunction of parametric vector optimization problems in banach spaces we provide inner outer estimates for evaluating the clarke circatangent coderivative of this multifunction in a broad class of conventional vector optimization problems in the presence of geometrical operator and finite and infinite functional constraints examples are given for analyzing and illustrating the obtained results

characterization of the law of the iterated logarithm in banach spaces on utilise une technique de randomisation gaussienne pour demontrer qu une variable aleatoire a valeurs dans un espace de banach satisfait la loi du logarithme itere sous une certaine condition necessaire et suffisante

almost sure behaviour of f valued random fields letx n n d be a field of independent random variables taking values in a semi normed measurable vector spacef for a broad class of fields n d of positive numbers the almost sure behaviour of k nxk n n d is studied the main result allows us to deduce some new and well known theorems for fields of independentf random variables from related results for fields of independent real random variables

subsmooth sets functional characterizations and related concepts prox regularity of a set poliquin rockafellar thibault 2000 or its global version proximal smoothness clarke stern wolenski 1995 plays an important role in variational analysis not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist c or the local uniqueness of the projection mapping but also because in the case where c is the epigraph of a locally lipschitz function it is equivalent to the weak convexity lower c 2 property of the function in this paper we provide an adapted geometrical concept called sub smoothness which permits an epigraphic characterization of the approximate convex functions or lower c 1 property subsmooth sets turn out to be naturally situated between the classes of prox regular and of nearly radial sets this latter class has been recently introduced by lewis in 2002 we hereby relate it to the mifflin semismooth functions

a note on regularity of sets and of distance functions in banach space abstract we show that in every infinite dimensional banach space there is a compact set c and x in c such that c is regular at x in the sense of nonsmooth analysis but such that the associated distance function d c is not regular at x and neither is d c

a numerical scheme for a class of sweeping processes the aim of this paper is to study a whole class of first order differential inclusions which fit into the framework of perturbed sweeping process by uniformly prox regular sets after obtaining well posedness results we propose a numerical scheme based on a prediction correction algorithm and we prove its convergence finally we apply these results to a problem coming from the modelling of crowd motion

prox regular sets and epigraphs in uniformly convex banach spaces various regularities and other properties we continue the study of prox regular sets that we began in a previous work in the setting of uniformly convex banach spaces endowed with a norm both uniformly smooth and uniformly convex e g l p w m p spaces we prove normal and tangential regularity properties for these sets and in particular the equality between mordukhovich and proximal normal cones we also compare in this setting the proximal normal cone with different holderian normal cones depending on the power types s q of moduli of smoothness and convexity of the norm in the case of sets that are epigraphs of functions we show that j primal lower regular functions have prox regular epigraphs and we compare these functions with poliquin s primal lower nice functions depending on the power types s q of the moduli the preservation of prox regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption a conical derivative formula for the metric projection mapping of prox regular sets is also established among other results of the paper it is proved that the attouch wets convergence preserves the uniform r prox regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

robust control design for nonlinear stochastic partial differential systems with poisson noise fuzzy implementation abstract this study addresses the robust h control design problem for nonlinear stochastic partial differential systems nspdss with poisson noise under the environment of random external disturbance in the spatio temporal domain for nspdss with poisson noise the robust h control design needs to solve a complex hamilton jacobi integral inequality hjii for robust control despite random external disturbance in general it is very difficult to solve the nonlinear partial differential hjii in order to simplify the design procedure a fuzzy stochastic partial differential system is proposed to approximate the nspds based on fuzzy interpolation approach then a fuzzy stochastic spatial state space model is developed to represent the fuzzy stochastic partial differential system via the semi discretization finite difference scheme and the kronecker product based on this model the robust h control design is proposed to achieve the robust control of nspdss via solving linear matrix inequalities lmis instead of an hjii the proposed robust fuzzy h controller has an efficient ability to attenuate the effect of spatio temporal external disturbance on the controlled output of nspdss from the area energy point of view finally a robust h control of the nervous system is given to confirm the control performance of the proposed robust control design method for nspdss with poisson noise

weak regularity of functions and sets in asplund spaces abstract in this paper we study a new concept of weak regularity of functions and sets in asplund spaces we show that this notion includes prox regular functions functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension we establish also that weak regularity is equivalent to mordukhovich regularity in finite dimension finally we give characterizations of the weak regularity of epi lipschitzian sets in terms of their local representations

some applications of isoperimetric methods to strong limit theorems for sums of independent random variables we develop several applications to almost sure limit theorems for sums of independent vector valued random variables of an isoperimetric inequality due to talagrand a general treatment of the classical laws of large numbers of kolmogorov and prokorov and laws of the iterated logarithm of kolmogorov and hartman and wintner is described new results as well as simpler new proofs of known ones illustrate the usefulness of isoperimetric methods in this context we show further how this approach can be used in the study of limit theorems for trimmed sums of independent and identically distributed random variables 1 introduction and theoretical statement isoperimetric methods already appeared in the study of strong limit theorems for sums of independent vector valued random variables through concentration inequalities mainly of gaussian type these inequalities are commonly used with the tool of randomization and applied conditionally in particular this was the basic idea of the main result of 24 which characterizes banach space valued random variables satisfying the classical hartman wintner strassen law of the iterated logarithm in a recent work 32 a new isoperimetric inequality for product measures was obtained and applied to the integrability of sums of independent vector valued random variables providing at the same time new perspectives in the study of strong limit theorems this inequality was indeed further applied in 25 to get extensions of kolmogorov s law of the iterated logarithm and prokorov s law of large numbers in banach space the purpose of this article is to demonstrate further the usefulness of this approach we study a general statement section 2 on sums of independent banach space valued random

subdifferentials of a minimum time function in banach spaces in general banach space setting we study the minimum time function determined by a closed convex set k and a closed set s this function is simply the usual minkowski function of k if s is the singleton consisting of the origin in particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function

limiting subdifferentials of perturbed distance functions in banach spaces abstract we explore in arbitrary banach spaces the frechet type e subdifferentials and the limiting subdifferentials for the perturbed distance function d s j determined by a closed subset s and a lower semicontinuous function j defined on s in particular upper and lower estimates for the frechet type e subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions j and s

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

infimal convolution and optimal time control problem iii minimal time projection set 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

linear regularity and ϕ regularity of nonconvex sets in this paper we discuss some sufficient conditions for the linear regularity and bounded linear regularity and their variations of finitely many closed not necessarily convex sets in a normed vector space the accompanying necessary conditions are also given in the setting of asplund spaces

subdifferentials of a minimal time function in normed spaces abstract in a general normed vector space we study the minimal time function determined by a differential inclusion where the set valued mapping involved has constant values of a bounded closed convex set u and by a closed target set s we show that proximal and frechet subdifferentials of a minimal time function are representable by virtue of corresponding normal cones of sublevel sets of the function and level or suplevel sets of the support function of u the known results in the literature require the set u to have the origin as an interior point or u be compact in particular if the set u is the unit closed ball the results obtained reduce to the subdifferential of the distance function defined by s