Reachable set analysis for linear systems with discrete and distributed delays

This paper is concerned with the reachable set analysis for linear systems with both discrete and distributed delays. By choosing appropriate Lyapunov-Krasovkii functionals, some sufficient conditions are established to ensure that all the states starting from the origin are bounded by an ellipsoid. It is shown that finding the smallest possible ellipsoid can be transformed into an optimization problem with matrix inequality constraints. In addition, the computational complexity is greatly reduced since much fewer variables are involved in the obtained results. These criteria are also extended to systems subject to polytopic uncertainties. It is shown that, in the absence of distributed delay, the obtained condition is less conservative than the existing ones.