Delay-Dependent Exponential Stability of Linear Distributed Parameter Systems

Exponential stability analysis via Lyapunov-Krasovskii method is extended to linear 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. Sufficient delay-dependent conditions for exponential stability are derived in the form of Linear Operator Inequalities (LOIs), where the decision variables are operators in the Hilbert space. Being applied to a heat equation and to a wave equation, these conditions are represented in terms of standard Linear Matrix Inequalities (LMIs). The proposed method is expected to provide effective tools for robust control of distributed parameter systems with time-delay.