We know that sometimes the controllability of  linearized control systems and fixed-point methods could imply the controllability of nonlinear control systems (KdV in noncritical cases, hyperbolique system and Schrödinger equations.
But when the  linearized control system becomes uncontrollable, e.g. KdV equation in critical cases, the power series expansion method introduced by Jean-Michel Coron will gives controllability of nonlinear control system though its linearized system is not controllable.
If control problem is an open problem, then stabilization problem should be a closed problem. It is in general more difficult than control problems, so far few advance has been made.In this talk I will mainly talk about the stabilization of KdV by means of feedback laws.   More precisely,we study the exponential stabilization problem for a nonlinear KdV on bounded interval in cases where the linearized control system is not controllable. The system has Dirichlet boundary conditions at the end-points of the interval, a Neumann nonhomogeneous boundary condition at the right end-point which is the control. We build a class of time-varying feedback laws for which the solutions of the closed-loop systems with small initial data decay exponentially to 0.  It is a joint work with Jean-Michel Coron and Ivonne Rivas.
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