Sampled-data control of semilinear 1-d heat equations ?

A semilinear one-dimensional convection-diffusion equation with distributed control, coupled to the Dirichlet or to the mixed boundary conditions, is considered. A sampled-data controller design is developed, where the sampled-data (in time) measurements of the state are taken in a finite number of fixed sampling points in the spatial domain. Sufficient conditions for the exponential convergence of the state dynamics are derived in terms of Linear Matrix Inequalities (LMIs) depending on the bounds of the sampling intervals. The new method is based on the direct Lyapunov approach via Wirtinger's and Halanay's inequalities.