On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems
The authors consider time-varying linear implicit differential equations \[ E(t)\dot{x}(t)=A(t)x(t),\quad t\geq 0, \] which are strongly index one, i.e.\ the equation can be decoupled into an explicite differential equation and an algebraic equation, furthermore, certain transformation matrices are assumed to be bounded. For this class of equations the stability radius with respect to structured dynamic perturbations of the matrix \(A\) and the Bohl exponent are defined and studied. It follows that the zero solution of the above equation is exponentially stable if, and only if, the Bohl exponent is negative and it is shown that exponential stability is equivalent to the property that the inhomogeneous equation \[ E(t)\dot{x}(t)=A(t)x(t)+f(t),\quad t\geq 0, \] has a ``bounded-input-bounded-output'' property. Finally, the dependency of the stability radius and of the Bohl exponent on additive changes in the matrix \(A\) are studied. The authors obtain ``continuity''-results, i.e.\ sufficiently small disturbances in the matrix \(A\) will not change the stability radius and the Bohl exponent significantly.
scientific article; zbMATH DE number 5354957