On the stability of solutions of certain linear set differential equations

TL;DR

This paper introduces new convex geometric methods to analyze the stability of solutions to linear set differential equations, establishing conditions for asymptotic stability in specific cases.

Contribution

It develops novel approaches using convex geometry and mixed volumes to study stability of linear set differential equations, including periodic cases.

Findings

Proved stability of program solutions with stable operators.

Established asymptotic stability conditions for equations with periodic operators in 2D.

Utilized convex geometry and homothety group actions in stability analysis.

Abstract

New approaches to the study of stability of solutions of Set Differential Equations (SDEs) based on convex geometry and the theory of mixed volumes were proposed. The stability of the forms of program solutions of linear SDEs with a stable operator was proved. We consider the orbit of the action of homotheties group on the space of nonempty convex compacts ($\conv\Bbb R^n$) as the form of a convex compact. For equations with periodic operator in the two-dimensional space the asymptotic stability conditions were established.