Carleman estimate and null controllability of a fourth order parabolic equation in dimension \(N\geq 2\) (Q1633075)

The authors consider a fourth order parabolic equation in a bounded smooth domain \(\Omega\) with homogeneous Dirichlet boundary conditions on the solution and the Laplacian of the solution. The main result of the work is establishing a Carleman inequality with observation in an arbitrarily small open set \(\omega\subset\Omega\), which leads to a null controllability result at any time \(T>0\) for the associated control system with a control function acting through \(\omega\). Much attention in the work is paid to analysis of known results in the area of existence and uniqueness of solutions to a fourth order parabolic equation, as well as to issues of stability of such equations. It is noted that Carleman estimates for these equations in a domain with dimension of two or greater are first established in this work. They were known previously for a 1-dimensional domain only.