Constructing Lyapunov-Krasovskii Functionals for Linear Time Delay Systems

We present an algorithmic methodology for constructing Lyapunov-Krasovskii (L-K) functionals for linear time-delay systems, using the sum of squares decomposition of multivariate polynomials to solve the related infinite dimensional Linear Matrix Inequalities (LMIs). The resulting functionals retain the structure of the complete L-K functional and yield conditions that approach the true delay-dependent stability bounds. The method can also be used to construct parameter-dependent L-K functionals for certifying stability under parametric uncertainty.