On data dependence of stability domains, exponential stability and stability radii for implicit linear dynamic equations (Q305699)

In this paper, the authors consider the following dynamic equation on a given time scale \(T\) \[ Ax^{\Delta}(t)=Bx(t),\quad x(s)=x_\theta,\quad s\in T_{t_\theta},\tag{1} \] where \(A,B\in\mathbb C^{m\times m}\), \(x(t)\in\mathbb C^m\), \(t\geq s\), \(t\in T_{t_\theta}\). The authors prove that (1.1) is uniformly exponentially stable if and only if \(\mu\) belongs to the domain of stability of \(T\), where \(\det(\mu A-B)=0\). Moreover using the perturbation of (1) as \[ \tilde{A}x^{\Delta}(t)=\tilde{B}x(t) \] they obtain the complex stability radius of (1.1) under \[ [\tilde{A},\tilde{B}]=[A,B]+D\Sigma E, \] where \(D\in\mathbb C^{m\times l}\), \(E\in\mathbb C^{q\times 2m}\) and \(\Sigma\in\mathbb C^{l\times q}\).