Sampled-data finite-dimensional boundary control of 1D parabolic PDEs under point measurement via a novel ISS Halanay's inequality

Recently, finite-dimensional observer-based controllers were introduced for 1D parabolic PDEs via the modal decomposition method. In the present paper we suggest a sampled-data implementation of a finite-dimensional boundary controller for 1D parabolic PDEs under discrete-time point measurement. We consider the heat equation under boundary actuation and point (either in-domain or boundary) measurement. In order to manage with point measurement, we employ dynamic extension and prove H¹-stability. Due to dynamic extension, which leads to proportional–integral controller, we suggest a sampled-data implementation of the controller via a generalized hold device. We take into account the quantization effect that leads to a disturbed closed-loop system and input-to-state stability (ISS) analysis. We use Wirtinger-based piecewise continuous in time Lyapunov functionals which compensate sampling in the finite-dimensional state and lead to the simplest efficient stability conditions for ODEs. To compensate sampling in the infinite-dimensional tail, we introduce a novel form of Halanay's inequality for ISS, which is appropriate for functions with jump discontinuities that do not grow in the jumps. Numerical examples demonstrate the efficiency of our method.