The LMI Approach in an Infinite-Dimensional Setting
Extended via the Lyapunov–Krasovskii method to linear time-delay systems (LTDS), the linear matrix inequality (LMI) approach has long been recognized as a powerful analysis tool of such systems. In this chapter, this approach is extended to the stability analysis of LTDSs evolving in a Hilbert space. The operator acting on the delayed state is supposed to be bounded. The system delay is unknown and time-varying, with an a priori given upper bound on the delay. Sufficient exponential stability conditions are derived in the form of linear operator inequalities, where the decision variables are operators in the Hilbert space. When applied to a heat equation and to a wave equation, these conditions are reduced to standard LMIs.
KeywordsInfinite-dimensional system Time-delay system Distributed parameter system Exponential stability Lyapunov–Krasovskii functional LMI
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