On the connected stability of the large-scale systems of dynamical equations on the time scale in critical cases
The authors provide a sufficient condition for the trivial solution of the so-called large-scale system \[ x^\Delta (t) = F(t,x), \quad\quad F(t,0 ) = 0 \] to be connectedly stable. Connected stability means that the trivial solution is asymptotically stable for all modified right hand sides \(F(t,Ex)\) with \(E\) a correlation matrix, that is, it has only entries in \(\{0,1\}\). The main assumption consists of the critical case, i.e., the matrix exponential functions and its inverse for the linearized equation are bounded. The long and technically involved proof of the main theorem is based on Lyapunov's direct method.
scientific article; zbMATH DE number 6110722