Exact boundary controllability for the linear Korteweg-de Vries equation – a numerical study.

*(English)*Zbl 0919.93039The author recalls that exact boundary controllability for the Korteweg-de Vries (K-deV) equation was established in his earlier paper [ESAIM, Control Optim. Calc. Var. 2, 33-55 (1997; Zbl 0873.93008)] using the Hilbert Uniqueness Method (HUM). The aim of this paper is to illustrate his results by numerical calculations. But he does more than that. He starts with the well-known dispersive partial differential equation, which may be regarded as a simple form of the K-deV equation, except for the term \(y_x\), which according to the author helps in modeling water waves in a uniform channel. The author raises the question: Does there exist an \(L^2[0, T]\) function \(h(t)\) such that
\[
y_x+ y_t+ yy_x+ y_{xxx}= 0,\;y(t,0)= y(t,\ell)= 0,\;y(0,x)= y_0(x),\;y_x(t,\ell)= h(t)?
\]
It turns out that the additional term \(y_x\) helps to establish exact boundary controllability, provided the term \(yy_x\) is ignored and the length \(\ell\) is sufficiently small. Otherwise, the answer to this question is not known.

The reviewer comments that if the length \(\ell\) is very large the answer is obviously “NO” irrespective of either presence or absence of the nonlinear term, or of any other term.

The author proceeds to describe a numerical algorithm based on the HUM. For convenience, he changes the spatial domain from \([0,\ell]\) to \([-1,+1]\), and after integration by parts derives the equation: \[ d/dt\langle y,\phi\rangle+ \langle y_x, (\phi_{xx}+ \phi)\rangle+ y_x(t, -1) \phi_x(-1)= h(t) \phi_x(1). \] The approximate solution is sought in the space of polynomials of known maximum degree. The time is discretized, and an implicit scheme is carried out. Now, the HUM is applied. The homogeneous Cauchy backward problem (in variable \(u(x,t)\)) is solved numerically. After \(u(t,1)\) is found, the solution of the boundary value forward problem is carried out. Thus the implicit scheme is used twice. First, backward in time for \(u(x,t)\) with \(h= 0\), then forward for \(y(x,t)\) with \(h= u_x(.,1)\). Several pages of graphics illustrate the outcomes of numerical experiments, which use collocation pseudo-spectral techniques.

The reviewer comments that if the length \(\ell\) is very large the answer is obviously “NO” irrespective of either presence or absence of the nonlinear term, or of any other term.

The author proceeds to describe a numerical algorithm based on the HUM. For convenience, he changes the spatial domain from \([0,\ell]\) to \([-1,+1]\), and after integration by parts derives the equation: \[ d/dt\langle y,\phi\rangle+ \langle y_x, (\phi_{xx}+ \phi)\rangle+ y_x(t, -1) \phi_x(-1)= h(t) \phi_x(1). \] The approximate solution is sought in the space of polynomials of known maximum degree. The time is discretized, and an implicit scheme is carried out. Now, the HUM is applied. The homogeneous Cauchy backward problem (in variable \(u(x,t)\)) is solved numerically. After \(u(t,1)\) is found, the solution of the boundary value forward problem is carried out. Thus the implicit scheme is used twice. First, backward in time for \(u(x,t)\) with \(h= 0\), then forward for \(y(x,t)\) with \(h= u_x(.,1)\). Several pages of graphics illustrate the outcomes of numerical experiments, which use collocation pseudo-spectral techniques.

Reviewer: Vadim Komkov (Florida)

##### MSC:

93C20 | Control/observation systems governed by partial differential equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

93B05 | Controllability |