Constrained control of 1-D parabolic PDEs using sampled in space sensing and actuation

We consider distributed control of a class of 1-D parabolic PDEs under distributed in-domain point actuation and measurements in the presence of control constraints. This class includes unstable diffusion-reaction equations as well as stable Burgers’ equations, where we aim to locally improve the convergence. We suggest an observer-based control law that employs the averaged values of the observer state. This allows to regionally stabilize the system. We derive linear matrix inequalities (LMIs) conditions that provide an estimate on the set of initial conditions starting from which the state trajectories of the system are exponentially converging to zero. A numerical example validates the efficiency of the method.