( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ]n ( ) ( ) [ ]n

(5) g ( x,
y,
u,
v)
= a( x) f ( y,
u) + C( y,
u,
v)
a
( x)
=
x
β
β
2
4
β β 1
+ β
3
+
x −
5
C (y,u,v) = δ1V1
−δ
2u
−δ
3
y
⎛ θ
4
V1(x,y,u) =
1u 2
y
3
f ( y,
u)
5
x ⎟ ⎞
θ + θ +
⎜θ
+
⎝ θ − ⎠
where A: α1... α
6;
β1...
β
5;
δ1...
δ
4;
θ1...
θ
5;a,
b are given constants.
The experimental data show that the state variables x, y must fall within preset limits.
(6)
y ∈
x ∈
[ y1,
y2
] = [ 30,110]
[ d ( y) , d ( y)
]
1
2
Other limits can also be set but the constants used to define the functions from (5) may
change.
As it can be observed, the functions defining system (4) are generally quite
complicated but the problem is now clearly formulated. It may also be noticed that (4)
is a command system whose data are non-differentiable (but are Liepschtesian), which
requires methods of the non-differentiable analysis that developed a lot recently.
Furthermore, the functions from (5) that define system (4) have a structure of stratified
functions and therefore the methods from (3) might apply.
Each of the optimisation problems presented above consists in the minimisation of a
cost functional defined as follows:
The problem of the minimum period is formulated as follows:
Given yF ∈ (y1, y2], for any
(x0,y0) ∈X0 = {( x, y) / y ∈ [ y1 , y2
); x ∈[ d1( y) , d
2
( y)
] } determine ~ t > F
0 and
2
( u~
(.), v ~ (.) ):[ 0, t
F
] → IR +
so that the system of differential equations from(4)
admits a solution ( ~ x (.),
~ y (.))
defined on [0,tF] that verifies the restrictions from (4) and
~ x t , ~ y t ∈ X ∀t
∈ 0,
~
t , y
~
t = y
(7) ( ( ) ( )) 0
[
F
] (
F
)
F
and which minimises the functional; C (u(.), v(.)) defined by:
t
(8) C (u(.), v(.)) = tF =
∫ F
dt
0
in the class of all the admitted commands having these properties.
In order to calculate the maximal amount of protein produced during a given period,
T>0 one must minimise functional C (.,.) defined by:
(9) C (u(.),v(.)) = -x(T) for each: