The central exponent and asymptotic stability of linear differential algebraic equations of index~1
The author considers perturbed time-varying linear differential algebraic equations (DAEs) of the form \[ A(t)\dot{x}(t) + B(t)x(t) + f(t,x(t))=0,\quad t\geq 0, \] where \(A\), \(B\) are bounded and continuous matrix-valued functions and \(f\) is a nonlinear perturbation. Throughout the paper it is assumed that the DAE has index one which -- roughly speaking -- means that the algebraic conditions can be solved and the DAE can be reduced to an ordinary differential equation (ODE). First, the central exponent of the unperturbed DAE (i.e.\ \(f\equiv 0\)) is defined. The central exponent can be viewed as an exponential growth rate of the solutions of the unperturbed DAE. It is then shown that the central exponent of the unperturbed DAE is equal to the central exponent of the inherent ODE. Finally, the influence of the perturbation on the central exponent of the DAE is studied and sufficient conditions for the perturbation \(f\) are given which ensure that the solution of the perturbed DAE is exponentially stable. The paper is well structured and nicely written.
scientific article; zbMATH DE number 5120080