Lack of exact controllability of a higher-order BBM system

The two-way propagation of a certain class of long-crested water waves is governed approximately by systems of equations of Boussinesq type. These equations have been put forward in various forms by many authors and their higher-order generalizations arise when modelling the propagation of waves on large lakes, oceans and in other contexts. Considered here is a class of such systems which couple two higher-order Benjamin–Bona–Mahony type equations. Our aim is to investigate the controllability properties of the linearized model posed on a bounded interval. More precisely, we study whether the solutions can be driven to a given state at a given final time by means of controls acting on the right endpoint of the interval. We show that the model is approximately controllable but not spectrally controllable. This means that any state can be steered arbitrarily close to another state, but no finite linear combination of eigenfunctions, other than zero, can be steered to zero. Our proofs rely strongly on a careful spectral analysis of the operator associated with the state equations.

Keywords

higher-order Boussinesq system, controllability, Fourier expansion, nonharmonic analysis

2010 Mathematics Subject Classification

35Q53, 46B15, 93B05, 93C20

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The first-named author was supported by Capes and CNPq (Brazil). The second-named author was partially supported by CNPq (Brazil).

Received 10 January 2021

Received revised 28 October 2021

Accepted 13 November 2021

Published 26 May 2022