Diagonalization and stability of multi-time-scale singularly perturbed linear systems
The paper deals with the system \(\dot x=A_ 0(t)x+A_{01}(t)y+A_{02}(t)z\), \(\epsilon \dot y=A_{10}(t)x+A_{11}(t)y+A_{12}(t)z\), \(\mu \dot z=A_{20}(t)x+A_{21}(t)y+A_{22}(t)z,\) where \(\epsilon\) and \(\mu\) are small parameters such that \(\lim_{\epsilon \to 0}(\mu /\epsilon)=0\). A transformation (originally proposed by Chang for one-parameter systems) is developed which results in transforming the system in a diagonal form according to the three time scales. This decomposition provides a basis for qualitative and asymptotic analysis of such singularly perturbed systems. In the paper it is used for studying the stability of the perturbed system assuming that the reduced system (with \(\epsilon =\mu =0)\) is exponentially stable.
scientific article; zbMATH DE number 3888100