Interval matrix differential equations

TL;DR

This paper introduces a method for solving matrix differential equations with uncertain matrix coefficients constrained within a set, enabling the analysis of continuous-time imprecise Markov chains through approximate solutions and error estimation.

Contribution

It defines an exponential of a set of matrices and provides an approximate solution method with error bounds for matrix differential equations with set-valued coefficients.

Findings

Defined an exponential of a set of matrices.

Developed an approximate solution method with error estimation.

Applicable to continuous-time imprecise Markov chains.

Abstract

The matrix differential equation $x'(t) = Q(t)x(t), x(0) = x_0$ is considered in the case where $Q(t)$ is an unspecified matrix function of time, with the only constraint that $Q(t)\in \mset$ for every $t$, where $\mset$ is a prescribed closed and convex set of matrices. We provide the solution of the generalised equation by defining an exponential of a set of matrices. Although the defintion is not directly applicable to calculate the solutions, we provide an approximate method along with the estimation of maximal possible error. In particular, the method allows estimating continuous time imprecise Markov chains.