On stability of linear parabolic distributed parameter systems with time-varying delays

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 known upper bound on the delay derivative. Sufficient delay-independent/delay-dependent stability conditions are derived in the form of linear operator inequalities, where the decision variables are operators in the Hilbert space. Being applied to a heat equation, these conditions are represented in terms of standard linear matrix inequalities.