Nonlocal balance equations with parameters in the space of signed measures. (1st January 2022)

• Record Type: Journal Article
• Authors: Pogodaev, N. I.; Staritsyn, M. V.
• Is Part Of: Sbornik. Volume 213:Number 1(2022)
• Page Start: 63
• Page End: 87
• Publication Date: 2022-01-01
• Subjects: 35R06 -- 49J20 -- 49J27; 37N25; nonlocal balance equations -- signed measures -- dynamical systems in measure spaces -- Kantorovich-Rubinstein distance; Mathematics -- Periodicals; 510.5
• DOI: 10.1070/SM9516
• Languages: English
• ISSNs: 1064-5616

Abstract

Abstract: A parametric family of nonlocal balance equations in the space of signed measures is studied. Under assumptions that cover a number of known conceptual models we establish the existence of the solution, its uniqueness and continuous dependence on the parameter and the initial distribution. Several corollaries of this theorem, which are useful for control theory, are discussed. In particular, this theorem yields the limit in the mean field of a system of ordinary differential equations, the existence of the optimal control for an assembly of trajectories, Trotter's formula for the product of semigroups of the corresponding operators, and the existence of a solution to a differential inclusion in the space of signed measures. Bibliography: 33 titles.