Stefan Rolewicz

On stability of linear time-varying infinite-dimensional discrete-time systems

The asymptotic behaviour of linear time-varying infinite-dimensional discrete-lime systems is considered. The introduced notions are: weak power equistability, power equistability, uniform power equistability, l p -equistability, uniform / l p -equistability and l p(x) -equistability. It is shown that they are identical. A generalization of the concept of spectral radius of a single operator is also proposed. It is proven that any time-varying system is uniformly power equistable if and only if the generalized spectral radius of the sequence of the operators which define the system considered is less than one.

Metric characterizations of upper semicontinuity

Abstract We characterize upper semicontinuity of multifunctions in terms of upper Hausdorff semicontinuity, measure of non compactness and active boundary. The results are applicable in optimization, theory of best approximation and in metrizability theory.

Exact Penalties for Local Minima

We provide a sufficient condition for the exact equivalence of constrained minimization problems and the minimization of associated generalized Lagrangians with respect to a perturbing class

On drop property for convex sets

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On the coincidence of some subdifferentials in the class of α(.)-paraconvex functions

. Exact equivalence amounts to equality of the sets of local solutions restricted to some region. The sufficient condition is expressed in terms of certain semicontinuity properties of objective functions and constraint multifunctions; for Banach spaces it becomes local controllability. The requirement is made more specific for mathematical programming.In this context we discuss properties of inner derivatives and approximations of multifunctions and we present a considerable extension of the Lusternik theorem.

A characterization of semicontinuity-preserving multifunctions

Let (X, ‖ · ‖) be a real Banach space. Let C be a closed convex set in X. By a drop D(x, C) determined by a point x ∈ X, x / ∈ C, we shall mean the convex hull of the set {x} ∪ C. We say that C has the drop property if C 6= X and if for every nonvoid closed set A disjoint with C, there exists a point a ∈ A such that D(a, C) ∩ A = {a}. For a given C a sequence {xn} in X will be called a stream if xn+1 ∈ D(xn, C) \ C (cf. [6]). When the set A has a positive distance from C, a variety of “Drop theorems” has been obtained in [1, 2, 3] and [8]. If C is the closed unit ball and has the drop property then we say that the norm ‖ · ‖ has the drop property [9]. Norms with the drop property have been investigated in papers [4, 6] and [9]. The drop property for closed bounded sets has been considered in [5]. There was proved that a bounded closed convex symmetric set having the drop property is compact or has a nonempty interior. We shall prove this theorem without assumptions on boundness and symmetry of sets under consideration. The Kuratowski measure of noncompactness of a set G in a Banach space X is the infimum α(G) of those ε > 0 for which there is a covering of G by a finite number of sets of diameter less than ε. For a closed convex set C denote by F (C) the set of all linear continuous functionals f ∈ X, f 6= 0, which are bounded above on C. For f ∈ F (C) and δ > 0 put

On nearly uniformly convex sets

In the paper an equivalence of Clarke, Dini, α(.)-subgradients and local α(.)-subgradients for strongly α(.)-paraconvex functions is proved

On Stability of Linear Time-Varying Infinite Dimensional Systems

Abstract We give a necessary and sufficient condition on a multifunction Γ that the function inf uϵΓ ( x ) f ( u ) be lower semicontinuous for all f from an arbitrary class F . The condition is then concretized for important special classes F .

ON UNCONDITIONAL CONVERGENCE OF LINEAR OPERATORS

The notion of nearly uniformly convex Banach spaces was introduced by Huff [2] and independently by Goebel and Sekowski [1]. In [7] it was shown that in a certain way it is a uniformization of drop property for norms. In [3] the authors considered drop property for convex sets. In the present paper they investigate nearly uniformly convex sets which need not be balls in Banach spaces. Let (X, ‖ · ‖) be a real Banach space. The norm (or the closed unit ball B) is said to be nearly uniformly convex (NUC) [2] if one of the following conditions is satisfied:

On Optimal Observability of Lipschitz Systems

An overview of recent results on the problem of stability of linear time-varying discrete-time and continuous-time systems is given. Particular attention is devoted to the relations between stability of a continuous-time system and its (appropriately defined) discretization.