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The shallow-water equations do not depend neither on the orientation nor on the origin of the frame in which the coordinates are expressed: they are invariant under the action of
, the Special Euclidean group of isometries of the plane
. Consequently, functions
and
must be invariant under the action of
. Symmetries have been very recently introduced for observer design in [1,40] for engineering problems. The aim of this work is to consider correction terms that respect the underlying physics of the system.
To find the scalar term
, we use the standard result (see e.g. [98]), which states that any scalar differential operator invariant by rotation and translation writes
, where
is a polynomial and
is the Laplacian. By considering the invariance by rotation for the vectorial velocity [87], we get the following family of scalar terms:
|
(3.52) |
where
and
are scalar polynomials in
, and
is an integral term defined below. More precisely,
|
(3.53) |
where
and
are smooth scalar functions such that
|
(3.54) |
For the vectorial correction term
, we use the vectorial counterpart:
|
(3.55) |
where
and
are polynomials in
, like
and
.
Let us now find the integral terms
and
that are invariant by rotation and translation. They can be expressed as a convolution between the previous invariant differential terms and a two-dimensional kernel
. The previous terms being invariant by rotation, the value of the kernel should not depend on a particular direction, and so
must be a function of the invariant
. The integral correction terms write:
|
(3.56) |
|
(3.57) |
where
and
are defined like
and
.
The support of
(resp.
) is a subset of
. Its characteristic size defines a zone in which it is significant to correct the estimation with the measurements. The integral formulation is actually quite general: if
and
are set equal to Dirac functions, one obtains the differential terms.
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