On the controllability of a Boussinesq system for two-way propagation of dispersive waves

In this paper, we are concerned with a Boussinesq system of Benjamin–Bona–Mahony type equation, posed on a bounded interval, modeling the two-way propagation of surface waves in a uniform horizontal channel filled with an irrotational, incompressible and inviscid liquid under the influence of gravit...

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Bibliographic Details
Published in:Journal of evolution equations 2020-06, Vol.20 (2), p.607-630
Main Authors: Bautista, George J., Pazoto, Ademir F.
Format: Article
Language:English
Subjects:
Analysis
Boussinesq equations
Computational fluid dynamics
Controllability
Eigenvectors
Equations of state
Mathematics
Mathematics and Statistics
Stability
Surface waves
Wave propagation
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Summary:In this paper, we are concerned with a Boussinesq system of Benjamin–Bona–Mahony type equation, posed on a bounded interval, modeling the two-way propagation of surface waves in a uniform horizontal channel filled with an irrotational, incompressible and inviscid liquid under the influence of gravitation. The main focus is on the boundary controllability property, which corresponds to the question of whether the solutions can be driven to a given state at a given final time by means of controls acting at one endpoint of the interval. We first show that the system is not spectrally controllable. This means that no finite linear combination of eigenfunctions associated with the state equations, other than zero, can be steered to zero. Although the system is not spectrally controllable, it can be shown that it is approximately controllable, i.e., any state can be steered arbitrarily close to another state.
ISSN:1424-3199
1424-3202
DOI:10.1007/s00028-019-00541-5