On Stability of Linear Retarded Distributed Parameter Systems of Parabolic Type

In this chapter the stability analysis via Lyapunov-Krasovskii method is extended to linear parabolic time-delay systems in a Hilbert space. The operator acting on the delayed state is supposed to be bounded. The system delay is admitted to be unknown and time-varying with an a priori given upper bound on the delay derivative, which is less than 1. Sufficient delay-independent asymptotic stability conditions are derived. These conditions are given in the form of Linear Operator Inequalities (LOIs), where the decision variables are operators in the Hilbert space. Being applied to a heat equation with the Dirichlet boundary conditions, these LOIs are represented in terms of standard Linear Matrix Inequalities (LMIs).