Exact boundary controllability of the Korteweg-de Vries equation

We consider boundary control of the distributed parameter system described by the Korteweg-de Vries (KdV) equation posed on a finite interval alpha less than or equal to x less than or equal to beta: [GRAPHICS] for t greater than or equal to 0. It is shown that by choosing appropriate control inputs (h(j)(t)); (j = 1; 2; 3), one can (always guide the system [asterisk] from a given initial state phi is an element of H-s (alpha, beta) to a given terminal state psi is an element of Hs (alpha, beta) in the time period [0; T] so long as phi and psi satisfy \\phi(.)-w(., 0)\\(Hs(alpha, beta)) less than or equal to delta and \\psi(.) - w(., T)\\(Hs(alpha,beta)) less than or equal to delta for some delta > 0 independent of phi and psi, where s greater than or equal to 0 and w = w(x; t) is a given smooth solution of the KdV equation. This exact boundary controllability is established by considering a related initial value control problem of the KdV equation posed on the whole line R. Various recently discovered smoothing properties of the KdV equation have played important roles in our approach.