Exact boundary controllability of a nonlinear KdV equation with critical lengths
The Korteweg-de Vries (KdV) equation originally described a weakly nonlinear propagation of a long water wave in a channel of constant cross-section. Early analysis of this problem can be found in an article of \textit{R. M. Miura}, \textit{C. S. Gardner} and \textit{M. D. Kruskal} [J. Math. Phys. 9, 1204--1209 (1968; Zbl 0283.35019)] and in an excellent survey by \textit{R. M. Miura} [SIAM Rev. 18, 412--459 (1976; Zbl 0333.35021)]. The authors start with the following widely used form of the KdV equation: \(y_t+ y_x+ y_{xxx}+ yy_x= 0\), with \(y(t,0)= y(t,L)= 0\). \textit{L. Rosier} pointed out [SIAM J. Control Optimization 39, No. 2, 331--351 (2000; Zbl 0966.93055)] that the linear system obtained by omitting the term \(yy_x\) is not controllable in the vicinity of the origin. Difficulty arose when the length was the critical length, and also some directions were ``missed'' in the linearized control system. The authors start by reviewing results of Rosier. They follow Rosier's arguments and study in great detail the controllability of the linearized system. They find that the missing directions, which are: \(1-\cos x\) and \(-(1-\cos x)\), can be taken care in the nonlinear equation. This is not as easy as it sounds. The article is 31 pages long, and there are several pages of just detailed computations. But the order of presentation leaves much to be desired. On page 394, the authors offer a proof of Lemma 11. This proof takes four pages of intricate computation, which starts without any explanation of what it is they are computing. But no matter how long one searches, there is no statement of Lemma 11. There are Lemmas 12, 16, 17, 22, but no sign of Lemma 11. The numbering of propositions also is not truly perfect. However, the authors were kind to the readers on occasions. Proposition 18 and a corollary were followed by immediate proof. The reviewer is grateful for small mercies. In an Appendix, the authors prove existence and uniqueness of solutions to the Cauchy problem for the nonlinear KdV equation.
scientific article; zbMATH DE number 2102880
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