Stability of differential-algebraic equations under uncertainty
In this paper, the authors consider the robust stability of a linear constant-coefficient differential-algebraic equation (DAE) \[ Ax'(t)+Bx(t)=0,\quad t\geq 0, \tag{1} \] where $A$ and $B$ are given real $n\times n$ matrices and $x(t)$ is the unknown $n$-dimensional function. Here, it is assumed that $\mathrm{det}A=0$. First, the structure of DAE (1) is characterized by the notion of unsolvability index. Then, the authors consider perturbed systems \[ (A+\Theta _1)x'(t)+(B+\Theta _2)x(t)=0,\quad t\geq 0, \tag{2} \] and investigate the robust stability of DAE (1) under perturbations. The authors define a special class of structured perturbations which does not change the internal structure of DAE (1). Then, under perturbations belonging to this class, the real stability radius of DAE (1) is defined, which is actually the stability radius of the underlying ODE of (1). Finally, as a simple consequence of the corresponding result for ODEs, a formula of the stability radius is obtained.
scientific article; zbMATH DE number 6962874