On a Control Problem for a System of Implicit Differential Equations

Pub Date : 2023-11-23 DOI:10.1134/s0012266123090124

E. S. Zhukovskiy, I. D. Serova

引用次数: 0

Abstract

We consider the differential inclusion \(F(t,x,\dot {x})\ni 0 \) with the constraint \(\dot {x}(t)\in B(t) \), \(t\in [a, b]\), on the derivative of the unknown function, where \(F\) and \(B \) are set-valued mappings, \(F:[a,b]\times \mathbb {R}^n\times \mathbb {R}^n\times \mathbb {R }^m\rightrightarrows \mathbb {R}^k \) is superpositionally measurable, and \( B:[a,b]\rightrightarrows \mathbb {R}^n\) is measurable. In terms of the properties of ordered covering and the monotonicity of set-valued mappings acting in finite-dimensional spaces, for the Cauchy problem we obtain conditions for the existence and estimates of solutions as well as conditions for the existence of a solution with the smallest derivative. Based on these results, we study a control system of the form \(f(t,x,\dot {x},u)=0\), \(\dot {x}(t)\in B(t) \), \(u(t)\in U(t,x,\dot {x}) \), \(t\in [a,b]\).

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一类隐式微分方程组的控制问题

摘要考虑未知函数导数的微分包含\(F(t,x,\dot {x})\ni 0 \)和约束\(\dot {x}(t)\in B(t) \), \(t\in [a, b]\)，其中\(F\)和\(B \)是集值映射，\(F:[a,b]\times \mathbb {R}^n\times \mathbb {R}^n\times \mathbb {R }^m\rightrightarrows \mathbb {R}^k \)是叠加可测的，\( B:[a,b]\rightrightarrows \mathbb {R}^n\)是可测的。利用有限维空间中集值映射的有序覆盖性质和单调性，得到了柯西问题解的存在性、估计性和导数最小解的存在性条件。基于这些结果，我们研究了一种形式为\(f(t,x,\dot {x},u)=0\), \(\dot {x}(t)\in B(t) \), \(u(t)\in U(t,x,\dot {x}) \), \(t\in [a,b]\)的控制系统。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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