Robust sampled-data control of a class of semilinear parabolic systems

We develop sampled-data controllers for parabolic systems governed by uncertain semilinear diffusion equations with distributed control on a finite interval. Such systems are stabilizable by linear infinite-dimensional state-feedback controllers. For a realistic design, finite-dimensional realizations can be applied leading to local stability results. Here we suggest a sampled-data controller design, where the sampled-data (in time) measurements of the state are taken in a finite number of fixed sampling points in the spatial domain. It is assumed that the sampling intervals in time and in space are bounded. Our sampled-data static output feedback enters the equation through a finite number of shape functions (which are localized in the space) multiplied by the corresponding state measurements. It is piecewise-constant in time and it may possess an additional time-delay. The suggested controller can be implemented by a finite number of stationary sensors (providing discrete state measurements) and actuators and by zero-order hold devices. A direct Lyapunov method for the stability analysis of the resulting closed-loop system is developed, which is based on the application of Wirtinger's and Halanay's inequalities. Sufficient conditions for the exponential stabilization are derived in terms of Linear Matrix Inequalities (LMIs). By solving these LMIs, upper bounds on the sampling intervals that preserve the exponential stability and on the resulting decay rate can be found. The dual problem of observer design under sampled-data measurements is formulated, where the same LMIs can be used to verify the exponential stability of the error dynamics.