LOCAL NULL-CONTROLLABILITY OF A PARABOLIC SYSTEM WITH COUPLED NONLINEAR BOUNDARY CONDITIONS

In this article, we study the boundary local null-controllability of a one-dimensional parabolic system with coupled nonlinear boundary conditions and only one single control. The significant point is that the state components are interacting only at the boundary points in terms of some nonlinear functions. The control function is acting either through a mixed nonlinear boundary condition on the first component or through a Neumann condition on the second component. The results are slightly different in the two cases. To study the controllability properties, we first consider the associated linear systems where the boundary nonlinearities are linearized around (0, 0). The method of moments helps us to prove the controllability and obtain a suitable control cost namely Ce C/T for the linearized systems. Then applying the source term method developed in [25], followed by the Banach fixed point argument, we obtain the small-time local boundary null-controllability of the system