In finite-dimensional control theory, feedback control using controllability Gramian goes back to results by Kleinman and Lukes in 1968 
 and 1970. Some years before, R. Bass characterized controls of minimal norms also using controllability Gramians. The extension of these 
 results to infinite-dimensional systems has a long history. Surprisingly, only reversible infinite-dimensional systems have been considered 
 in those results. We shall present existing results in the literature and we shall characterize stabilizing controls of minimal norms for 
 parabolic systems. This is a joint work with S. Kesavan. Application to the stabilization of the Navier-Stokes equations will be given.