Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 205

dt ¼ X

N

n i

d

a ijk n j ð t Þn k ð t Þþ b ij n j ð t Þ ð 2 : 35 Þ

k ; j¼1

Putting the initial conditions as

n i ðÞ i ¼ 1 ð ; ...; N Þ ð 2 : 36 Þ

the reconstruction task of ʾ i (t) for the arbitrary time t ∈ [0,T] comes down to the

simple task of determining unknown coef cients based on the criterion:

( )

E ¼ X

X

M N

i¼1 n i ð t s Þn i ð t s Þ

2

½ = N = M ð 2 : 37 Þ

s¼1

where 0 t 1 t M T.
Coef cients {a ijk ,b ij ) can be evaluated as solution of the following optimization

task:

E 0 ¼ min

a ijk ; b ij ;n i ð 0 Þ

E ð 2 : 38 Þ

f g

There are many methods to solve this task. One of them is based on the
Bellman s dynamic programming method (Bellman and Roth 1966; Krapivin and

Kondratyev 2002; Krapivin 1969; Ram 2010). In this case search of minimal value
of function E comes to the dynamic programming task. Let us supposed that
coef

'

cients {a ijk ,b ij ) are functions of time:

n 1 ð t Þ
.
n N ð t Þ
a 111 ð t Þ
.
a NNN ð t Þ
b 11 ð t Þ
.
b NN ð t Þ

Y ð t Þ ¼ ð 2 : 39 Þ

We can consider that a ijk =a ikj . Then addicting the system ( 2.35 ) Cauchy
problem ( 2.35 ), ( 2.36 ) comes to the following task:

dY = dt ¼ G ðÞ ð 2 : 40 Þ