Exponential Stability and Relative Controllability of Nonsingular Delay Systems

In this paper we consider a delayed matrix exponential and use it to derive a representation of solutions to a linear nonsingular delay problem with permutable matrices. Also we present some sufficient conditions to guarantee that the trivial solution of such delay systems are exponentially stable. In addition using a delay Grammian matrix we present a criterion for a linear problem to be relatively controllable. Nonlinear problems are also discussed. Numerical examples are given to illustrate our theory.