This article is cited in 7 papers

Nonlocal balance equations with parameters in the space of signed measures

N. I. Pogodaev^ab, M. V. Staritsyn<sup>b

a</sup> N. N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
^b Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk, Russia

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.

Keywords: nonlocal balance equations, signed measures, dynamical systems in measure spaces, Kantorovich-Rubinstein distance.

UDC: 517.955

MSC: Primary 35R06, 49J20, 49J27; Secondary 37N25

Received: 21.10.2020 and 19.04.2021

DOI: 10.4213/sm9516

 English version: Sbornik: Mathematics, 2022, 213:1, 63–87

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