Delayed H_∞ control of 2D diffusion systems under delayed pointlike measurements

Up to now, robust control of multi-dimensional diffusion systems was confined to averaged measurements. In this paper, we consider 2D diffusion systems with delayed pointlike measurements. A pointlike measurement is the state value averaged over a small subdomain that approximates its point value. The main novelty enabling the study of such measurements is a new inequality, which we call the reciprocally convex variation of Friedrich's inequality. It bounds the difference between a function and its point values in the L²-norm using the function's derivatives. Combining this result with a new Lyapunov–Krasovskii functional, which has a spatially-varying kernel, we solve the H_∞ control and filtering problems in the presence of time-varying input and output delays. We show that any 2D semilinear diffusion system with pointlike measurements can be stabilized by static output feedback applied through characteristic functions if the controller gain and number of sensors/actuators are large enough while the input and output delays are sufficiently small. The results are demonstrated on a 2D catalytic slab model.