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

Optimal control of some metabolism processes
R. Burlacu
University of Agricultural Sciences and Veterinary Medicine, Bucharest
During the recent years a collective from the Institute of Biology and Animal
Nutrition, Baloteti developed a set of mathematical models of the metabolism
processes for some categories of farm animals. These models describe the evolution of
some synthetic indicators of the process of metabolism (body proteins and lipids)
function of some characteristics of the given food (dietary protein and energy).
Thus, if we note:
Mn = the amount of dietary protein given at moment n;
Vm = the amount of dietary energy given at moment n;
Xn = the amount of body protein at moment n;
Zn = the amount of body lipids at moment n.
We may observe using the existing empirical formulas that these measures are linked
by relations such as:
~
⎪⎧
X
n
+ 1 = X
n
+ f ( xn
, zn
, un
)
⎨
⎪⎩ Z
n
+ 1 = zn
+ g~
( xn
, zn
, unVn
)
where:
Z ∈ [ ~ z , ~ z n 1 2
]; ∈ ~ ~
x
n
[ d1,
( zn ),
d
2
( z n
)]
V
n
∈[ u~
( xn
zn
) u~
1
,
3( xn
, zn
)]
V
n
∈[ v~
( xn
zn
un
), v ~
1
, ,
2
( xn
, zn
)]
~
~ ~
where the functions: f , g~ , u~ 1 , u~
2, v ~ 1, v ~
2,
d1,
d
2
have a rather complicated structure.
A discrete command system was thus developed in which the command variables
(inputs) were un, vn, and the state variables were xn, zn.
The early studies were conducted in order to validate this model by selecting various
strategies (commands (un, vn)) according to the practical experience and by calculating
the outputs (xn, zn) of the model. The experimental data were compared to the
calculated ones and a mean error of 3% was noticed (the standard mean error for this
field is 5%), which showed that the model simulated quite well the actual processes.
The following step was to formulate and solve problems of optimisation; to calculate
commands which to optimise some target functions. Following are the most common
targets among the many possibilities:
1) Produce a given total weight (function of xn, zn) within a minimal period
of time;
2) Produce a maximal amount of protein within a given period of time;
3) Minimize the cost of reaching the target weight at the set deadline.
The complexity of the functions used by the model and of the restrictions (the
commands must have values within the intervals depending on the previous outputs)

make it very difficult, if not impossible, to perform a full and rigorous study of these
problems within the discrete interval mentioned earlier.
On the other hand, some recent results on the dynamic programming within the
theory of the optimal control as well as the existence of performing software for
computer integration of the differential equations seem to offer great chances of
solving them within a continuous model.
The careful analysis of the real phenomena forming the background of the discrete
model yields grounds for the development of a continuous model of the same pattern.
Thus, if we note:
x(t) = the amount of body protein at moment t
z(t) = the amount of body lipids at moment t
u(t) = the amount of dietary protein given at moment t
v(t) = the amount of dietary energy given at moment t
then, these functions are linked by differential equations:
~
⎧x'(
t)
= f ( x( t) , z( t) , u( t)
)
(2) ⎨
⎩z'
( t) = g~
( x( t) , z( t) , u( t) , v( t)
)
with u (t) ∈[ u ~ ( x( t) z( t)
) u~
1
, ,
3( t) , z( t)
]
v(t) ∈[ v ~ ( x( t) , z( t) , u( t)
), v ~ ( x( t) , z( t)
)]
1
2
The analysis of ~ f , g~ , u~ ~
1 , u2, v ~ 1, v ~
2
functions shows that the system may be
considerably simplified if variable z(t) is replaced by variable y(t) = total body weight
at moment, given by:
(3) y( t)
x
=
( t) + z( t) + α + z( t)
1−α
1
3
α
2
where α
1
, α
2,
α
3
> 0 are given constants.
With the new variables (x(.),y(.)), which are more convenient as terminal conditions
and as criteria of optimisation too, the command system (2) becomes:
x ( ( ))
( ) [ ( ) ( )]
( ) [ ( ( )),
v ( y)
]
⎧ ′ = f y,
u t , u t ∈ u1
y , u3
y
(4) ⎨
⎩y′
= g( x,
y,
u( t) , v( t)
),
v t ∈ v1
x,
y,
u t
where functions ~ f , g~ , u~ ~
1 , u2, v ~ 1, v ~
2
are as following:
2
f(y,u) =
⎧b
⎨
⎩A;
( y u)
α
= α u −α
y ; u ∈ u ,( y) , u ( y)
6
,
4 5
u ∈
[
1 2
]
[ u ( y) , u ( y)
]
2
2
α
A
u
1
=
2 1
;
3
+
α
α
5 α6
( y) y ; u ( y) = u ( y) + u ( y) = ay b
4
4

(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:

(t0,x0,y0) ∈ E0 = {( , x,
y) / t ( 0, T ),
y ∈( y , y ),
x ∈[ d ( y) d ( y)
]}
t
1 2
1
,
2
of all admitted commands (u(.), v(.)) : [0,T] → IR 2 +
and the restrictions:
∈ in class υ (t0,x0,y0)
that verify the restrictions from (4)
(9) (t,x(t), y(t)) ∈E0 ∀ t ∈(t0,T)
In order to minimise the cost of producing the target weight within the set deadline,
T>0 one must minimise functional C (.,.) defined by:
(10) C(u(.), v(.)) = ( C u( t) C v( t)
)
t
∫ 1
+
2
dt.
0
where C1,C2 > 0 are the prices of the dietary protein and energy.
The next step to be taken subsequently is to use the results and models from the
literature to solve these problems of optimal control.
References
Burlacu Gh., R. Burlacu, I. Columbeanu, G. Alexandru
proceselor de metabolism la monogastrice
, 1987 -
Român de Biometrie la Academia R.S.R. –
1986 - Sufficient optimality condition for stratified optimal control
problems, SIAM J Cont and Opt 24 (), 675-695.
–
Whittemore C.T. 1983 - Agricultural Systems, p 159-186.