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.

中文翻译：

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带符号测度空间中参数的非局部平衡方程

研究了带符号测度空间中的非局部平衡方程的参数族。在涵盖许多已知概念模型的假设下，我们确定解的存在性、其唯一性以及对参数和初始分布的持续依赖性。讨论了该定理的几个对控制理论有用的推论。特别是，该定理得出常微分方程组平均场的极限、轨迹集合的最优控制的存在性、相应算子的半群乘积的 Trotter 公式以及解的存在性差异包含在已签署措施的空间中。参考书目：33 个标题。