Measure Differential Inclusions: Existence Results and Minimum Problems

Abstract

We focus on a very general problem in the theory of dynamic systems, namely that of studying measure differential inclusions with varying measures. The multifunction on the right hand side has compact non-necessarily convex values in a real Euclidean space and satisfies bounded variation hypotheses with respect to the Pompeiu excess (and not to the Hausdorff-Pompeiu distance, as usually in literature). This is possible due to the use of interesting selection principles for excess bounded variation set-valued mappings. Conditions for the minimization of a generic functional with respect to a family of measures generated by equiregulated left-continuous, nondecreasing functions and to associated solutions of the differential inclusion driven by these measures are deduced, under constraints only on the initial point of the trajectory.

A note on convergence results for varying interval valued multisubmeasures

“…In the theory of integration several generalizations have been formulated extending the Riemann and Lebesgue integrals (see for example [6-10, 26, 27, 33] and the references therein). In the present paper we continue the research started in [7,13,20,34] and originating from [24,27]. We provide sufficient conditions for obtaining a Monotone Convergence Theorem for a sequence of interval valued multifunction (G n ) n with respect to interval valued multisubmeasures M and we consider also the asympotic properties of G n dM n for a pair of sequences (G n , M n ) n of interval valued multifunctions and multisubmeasures providing a generalized version of the Monotone Convergence Theorem [22,28,31,41].…”

Set-valued problems under bounded variation assumptions involving the Hausdorff excess

“…the Hausdorff-Pompeiu distance or in [12] via the notion of uniform bounded ε-variation ( [16]). Besides, closure properties of the solution set were obtained in [11] using convergence results for Stieltjes integrals and minimality results were given in [13].…”

“…In particular, in a finite dimensional space X, existence results can be found for single-valued framework in [18] and for the set-valued setting in [32], [9], [13]. Moreover, a continuous dependence result (i.e.…”

“…Our idea is to improve some of such known results, firstly by imposing conditions related to the Hausdorff excess instead and secondly by working in a Banach space (note that under Hausdorff excess assumptions compactness features and minimality results were given in [13] in finite dimensional spaces). This is possible due to the works of V.V.…”

On Regulated Solutions of Impulsive Differential Equations with Variable Times

“…In this paper we study impulsive IVPs in the space G([a, b]) of regulated functions, which seems to be the natural space of solutions for impulsive problems (see [18][19][20]), and we investigate properties of solutions as elements of this space. This allows us to cover and extend earlier approaches.…”