Measure Differential Inclusions: Existence Results and Minimum Problems
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Measure Differential Inclusions: Existence Results and Minimum Problems Luisa Di Piazza1 · Valeria Marraffa1
· Bianca Satco2
Received: 30 December 2019 / Accepted: 27 September 2020 / © The Author(s) 2020
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 HausdorffPompeiu 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. Keywords Measure differential inclusion · Bounded variation · Pompeiu excess · Selection · Minimality condition Mathematics Subject Classification (2010) 26A45 · 34A60 · 28B20 · 26A42 · 49Kxx
Valeria Marraffa
[email protected] Luisa Di Piazza [email protected] Bianca Satco [email protected] 1
Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123, Palermo, Italy
2
Faculty of Electrical Engineering and Computer Science; Integrated Center for Research, Development and Innovation in Advanced Materials, Nanotechnologies, and Distributed Systems for Fabrication and Control (MANSiD), Stefan cel Mare University of Suceava, Universitatii 13, Suceava, Romania
L. Di Piazza et al.
1 Introduction In the dynamic of many systems in physics, engineering, biology or chemistry, one has to face the occurrence of discontinuities in the state, which can be seen as impulses. One way to mathematically describe such systems is offered by the theory of measure differential equations. On the other hand, in various situations (e.g. when a control is involved), it is more convenient to consider multivalued functions, i.e. differential inclusions driven by Borel measures (thus allowing a unified approach of differential or difference set-valued problems, of impulsive problems or even of dynamic inclusions on time scales [11, 19]). Usually in the literature concerning the theory of differential inclusions, the HausdorffPompeiu distance appears when writing the conditions imposed on the right-hand side. Relaxing the traditional hypotheses by using the Pompeiu excess (from the right or from the left) instead of Hausdorff-Pompeiu metric would be a consistent improvement. In the present work, we study non-convex measure differential inclusions dx(t) ∈ G(t, x(t))dμg (t), x(0) = x0
(1.1)
with x0 ∈ Rd , under excess bounded variation assumptions (inspired from [13], see also [12]) on the velocity set G(t, x(t)) and make use of interesting sele
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