According to the integral feature of the Dirac function
, distributed impulsive control protocol is described as
To perform the sampling, it becomes necessary to define the Dirac function
as a distribution as follows:
Here, [[delta].sub.[epsilon]] - [H'.sub.[epsilon]] is the univariate Dirac function
and [H.sub.[epsilon]] is the Heaviside function defined in (9) and (10), respectively, while g(I) is edge indicator function and spf(f) is locally computed SPF function defined in (11) and (14), respectively.
where [delta][??] is the Dirac function
, and n denotes the exterior normal to the boundary [partial derivative][OMEGA].
Hence, the dimension [[L.sup.-4]] of the Lagrangian density means that the dimension of the Dirac function
[psi] is [[L.sup.-3/2]].
The following are Dirac function
and Heaviside function, respectively:
where: [dm.sub.k] is the mass of droplet k, [H.sub.k] is the specific enthalpy of liquid fuel, r position vector and [delta] is the Dirac function
It can be observed that for [lambda] [right arrow] 0 we obtain the Dirac function
where [delta] is the univariate Dirac function
, H is the Heaviside function, [L.sub.a] ([[empty set].sub.[mu]])is the length of the zero level curve, [A.sub.g]([[empty set].sub.[mu]]) is used to speed up curve evolution which is the weighted area of the subregion, and g is the fuzzy edge indicator function:
The dimension of a Dirac function
is [[L.sup.-3/2]] and the dimension of the electromagnetic 4-potential is [[L.sup.-1]].
Here the inner product of the Hilbert space is based on the density of the Dirac function
Since the system is linear the response of the system on the Dirac function
was convoluted with a temporally Gaussian distributed laser radiation.