  In this paper, we analyze the instability of continuous-time Markov jump
linear systems. Although there exist several effective criteria for the
stability of Markov jump linear systems, there is a lack of methodologies for
verifying their instability. In this paper, we present a novel criterion for
the exponential mean instability of Markov jump linear systems. The main tool
of our analysis is an auxiliary Markov jump linear system, which results from
taking the Kronecker products of the given system matrices and a set of
appropriate matrix weights. We furthermore show that the problem of finding
matrix weights for tighter instability analysis can be transformed to the
spectral optimization of an affine matrix family, which can be efficiently
performed by gradient-based non-smooth optimization algorithms. We confirm the
effectiveness of the proposed methods by numerical examples.
