External Direct Sum Invariant Subspace and Decomposition of Coupled Differential-Difference Equations

This article discusses the invariant subspaces that are restricted to be external direct sums. Some existence conditions are presented that facilitate finding such invariant subspaces. This problem is related to the decomposition of coupled differential-difference equations, leading to the possibility of lowering the dimensions of coupled differential-difference equations. As has been well documented, lowering the dimension of coupled differential-difference equations can drastically reduce the computational time needed in stability analysis when a complete quadratic Lyapunov–Krasovskii functional is used. Most known ad hoc methods of reducing the order are special cases of this formulation.