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Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line. (English) Zbl 0966.93055

The author considers controllability of the Korteweg-de Vries equation in the form

u t +u x +uu x +u xxx =0,t0,xΩ,(1)

and of the linear equation

u t +u x +u xxx =0,(2)

obtained by dropping the nonlinear term uu x . The control is applied at the endpoint x=0. Up to this time no boundary controllability results were known for an unbounded domain, such as Ω=(0,). The present paper contains proofs filling this gap.

First, the author observes that approximate controllability is easy to prove for the linear equation (2) in L 2 (0,), but exact controllability proof runs into trouble because of lack of compactness. In fact, the author proves that in the linear case there exists u 0 L 2 (0,) such that with u t (0)=u 0 , for any T>0, u t=T 0, where u is the solution of (2). It should be noted that the spaces of functions to which the initial and final states belong are different. However, difficulties arising here are overcome by use of Holmgren’s uniqueness theorem. Following ideas of Fursikov and Immanuvilov in the study of boundary controllability of Burger’s equation, the author proves another version of Carleman’s global estimate. His final result, following his arguments about existence of continuous semigroups, determines that the exact boundary controllability exists provided the solutions are not required to be bounded in L (0,T,L 2 (0,)). He also shows that this result is false if boundedness is assumed.

This is a clever proof, using alternatively the nonlinear equation (1) and the linear equation (2), and overcoming several difficult obstacles.


MSC:
93C20Control systems governed by PDE
93B05Controllability
35Q53KdV-like (Korteweg-de Vries) equations