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Abstract. The exact boundary controllability of linear and nonlinear Korteweg-de Vries equation on bounded domains with various boundary conditions is studied. When boundary conditions bear on spatial derivatives up to order 2, the exact controllability result by Russell-Zhang is directly proved by means of Hilbert Uniqueness Method. When only the rst spatial derivative at the right e n d p o i n t is assumed to be controlled, a quite di erent analysis shows that exact controllability h o l d s too. From this last result we derive the exact boundary controllability for nonlinear KdV equation on bounded domains, for su ciently small initial and nal states.

This paper aims at completing an earlier work of Russell and Zhang [38] to study internal control problems for the distributed parameter system described by the Korteweg-de Vries equation on a periodic domain T. In [38], Russell and Zhang showed that the system is locally exactly controllable and locally exponentially stabilizable when the control acts on an arbitrary nonempty subdomain of T. In this paper, we show that the system is in fact globally exactly controllable and globally exponentially stabilizable. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of propagation of compactness and propagation of regularity in Bourgain spaces for solutions of the associated linear system. Then, using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate.