INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 15, Issue: 2, Month: April, Year: 2020
Article Number: 3859, doi.org/10.15837/ijccc.2020.2.3859

CCC Publications 

Optimal Control Applications in the Study of Production
Management

L. Popescu, N. D. Militaru, O. M. Mituca

Liviu Popescu*
Department of Statistics and Economic Informatics
Faculty of Economics and Business Administration, University of Craiova,
Al. I. Cuza st. no. 13, Craiova 200585, Romania
*Corresponding author: liviunew@yahoo.com

Nicolae Daniel Militaru
University of Craiova
Al. I. Cuza st. no. 13, Craiova 200585, Romania
Postdoctoral researcher
militarunicolaedaniel@yahoo.com

Ovidiu Mircea Mituca
Academy of Economic Studies
Piata Romana, no. 6, Bucharest 010374, Romania
PhD student
mircea.mituca@gmail.com

Abstract

A mathematical model for an economic problem of production management is proposed. The
continuous optimal control problem is solved, by using the Pontryagin Maximum Principle at the
level of a new space, called Lie algebroid. The controllability of the economic system is studied
by using Lie geometric methods and involves restrictions on the final stock quantities. Finally, a
numerical application is given.

Keywords: optimal control, automatic control, controllability, production management, Pon-
tryagin Maximum Principle, Lie algebra.

1 Introduction
The mathematical methods are applied successfully in optimal control theory with many appli-

cations in economics, business, finance or engineering. One of the motivations for this paper is the
study of some type of Lagrangian systems with external constraints. These systems have important
applications in many different areas as optimal control theory, econometrics, cybernetics, operational
research (Caputo [3], Feichtinger & Hartl [8], Liu [14], Sethi & Thompson [22], Weber, [23] ) or au-
tomatic control (Caruntu et al. [4], Chitsaz et al. [6], Dubins [7]). Also, in control theory, the Lie



doi.org/10.15837/ijccc.2020.2.3859 2

geometric methods have been applied by many authors, see for example Agrachev & Sachkov [1],
Brocket [2], Isidori [10], Jurdjevic [11], LaValle [13], Popescu [20].
In this paper we propose a mathematical model for a problem of production management, using the
optimal control techniques. We find the optimal solution with minimum costs of production and stor-
age applying the Pontrygin Maximum Principle (PMP) at the level of a new working space, called Lie
algebroid, which is a holonomic distribution of the tangent space. Moreover, it is provided that the
framework of Lie algebroids is more suitable than the tangent space in the study of control affine sys-
tems with holonomic distributions. The controllability of our economic system is studied by using Lie
geometric methods and applying the Frobenius theorem. Such type of problems including inventory
and production management are intensely studied. Thus, Chazal, Jouini & Tahraoui [5] studied the
production planning and inventories optimization, using a backward approach for the case of convex
storage cost. Gayon, Vercraene & Flapper [9] dealt with optimal control of a production inventory
system with two disposal options and product returns. Maccini, Moore & Schaller [15] studied the
inventory behavior with permanent sales stocks. Olsson [18] presented some modelling techniques for
base-stock inventory systems in the case of state dependent demand rates. Ortega & Lin [19] gave a
review on control theory applications to the production and inventory problems. Popescu [21] used
the framework of a driftless control affine system in the study of inventory and production problems.
The present paper is organized as follows. In the second section we present the known results about
Lie geometric methods in optimal control theory for control affine systems, including the controlla-
bility problems in the case of holonomic distributions. In section three, that contains the novelty of
the paper, we give an application of optimal control to a problem of production management and
show that the Hamilton-Jacobi-Bellman equations, provided by PMP on cotangent space, lead to a
system of differential equations, very difficult to solve. In order to find the optimal solution, we will
use a different way, applying the PMP in the framework of Lie algebroids. Next, we present only
the necessary notions about Lie algebroids (see Mackenzie [17] for more details) and the geometric
viewpoint of the optimal control. Using the relation between the Hamiltonian functions on dual Lie
algebroid and the cotangent space, we apply the PMP at the level of a Lie algebroid, which in this
case is a distribution on the tangent bundle. Next, we prove that the distribution generated by the
vector fields is integrable and it determines a foliation in the state space. It results that the proposed
economic system is not controllable, in the sense that we can only manufacture a certain final quantity
of products. In the last part of the section, we find the complete solution of the control system using
the PMP on Lie algebroids. Finally, a numerical example is given.

2 Theoretical basic of control affine systems
Let us consider M be a smooth and n-dimensional manifold. A continuous control system is given

by the differential equations, depending on some parameters

dxi(t)
dt

= fi(x(t),u(t)),

where x ∈ M represents the state of the system and u ∈ U ⊂ Rm represents the controls. Considering
x0 and x1 two points of M (two states of the system), then the optimal control problem consists of
finding the optimal trajectories of control system that connect x0 and x1 and minimizing the cost

min
∫ T

0
L(x(t),u(t))dt, x(0) = x0, x(T) = x1,

where the function L is the Lagrangian (energy, cost, time, distance, etc.). The most important
and powerful tool for studying the optimal solutions in control theory is PMP. It generates the dif-
ferential equations of first order, necessary for the optimal solutions. For each optimal trajectory,
c(t) = (x(t),u(t)), it offers a lift on the cotangent space (x(t),p(t)) satisfying Hamilton-Jacobi-Bellman
equations. The Hamiltonian function is given by H(x,p,u) = 〈p,f(x,u)〉−L(x,u), p ∈ T∗M, and it
results that the maximization condition with respect to u, given by

H(x(t),p(t),u(t)) = max
v

H(x(t),p(t),v),



doi.org/10.15837/ijccc.2020.2.3859 3

yields ∂H
∂u

= 0 (H is assumed to be smooth with respect to u). The extreme trajectories must satisfy
the following equations

ẋ =
∂H

∂p
, ṗ = −

∂H

∂x
. (1)

We recall that a control affine system has the form [2]

ẋ = X0(x) +
m∑
i=1

uiXi(x), (2)

where x = (x1, ...,xn) represent the local coordinates on the manifold M, u(t) = (u1(t), ...,um(t))
∈ U ⊂ Rm, m ≤ n and X0,X1...Xm are smooth vector fields on M. Usually, X0 is called the drift
vector field which describes the dynamics of the affine system in the absence of control. The vector
fields Xi, i = 1,m are called the input vector fields and the function u(t) represents the control. We
can say that the system is controllable if for any two states x0 and x1 of the system, there exists a
finite time T and a control u : [0,T] → U such that for x(t) with x(0) = x0, we have x(T) = x1. In
other words, the system is controllable if for any two states x0, x1, there exists a solution curve of
the system (2) which connects x0 to x1. Controllability is the ability to move a system from a given
initial state to any final state, in finite time, using some available controls. A distribution ∆ on M
is an application which assigns for every point in M a subspace of the tangent space at this point.
The distribution ∆ is called locally finitely generated if there is a set of vector fields {Xi}i=1,m (called
local generators of ∆) which spans ∆ , i.e. ∆(x) = span{X1(x), ...,Xm(x)}⊂ TxM. The distribution
∆ has dimension k if dim ∆(x) = k, for all points x in M. The Lie bracket of two vector fields is given
by [Xi,Xj](x) =

∂Xj
∂x

(x)Xi(x) − ∂Xi∂x (x)Xj(x), (
∂Xj
∂x

is the Jacobian matrix of Xj). A distribution ∆
on M is said to be involutive if, for any x ∈ M and Xi(x),Xj(x) ∈ ∆(x), we have [Xi,Xj] (x) ∈ ∆(x).
If the involutive distribution is generated by vector fields {Xi}i=1,m then it results

[Xi,Xj] (x) =
m∑
k=1

Lkij(x)Xk(x).

Therefore, we can say that each Lie bracket is expressed as a linear combination of the system vector
fields. We recall that a foliation {Sα}α∈A of M is a partition of the space M = ∪Sα of M in disjoint
and connected submanifolds Sα, called leaves. A distribution ∆ of constant dimension on M is called
holonomic (integrable) if there exists a foliation {Sα}α∈A on M whose tangent bundle is ∆, that is
TxS = ∆(x), where S is the leaf passing through x. Frobenius’s theorem says that if the distribution
∆ has constant dimension, then ∆ is integrable if and only if ∆ is involutive. The presence of the drift
X0 in the study of control affine systems complicates the question of controllability. In the following
we deal with the driftless control affine system in the form

ẋ =
m∑
i=1

uiXi(x). (3)

It is assumed that the vector fields Xi, i = 1,m, generate a distribution ∆ on the connected manifold
M so that the rank of ∆ is constant. If the distribution ∆ = span{X1,X2, ...,Xm} is holonomic
with constant rank, then [Xi,Xj] ∈ ∆ for every i 6= j and the system is not controllable. Using the
Frobenius theorem it results that the distribution ∆ is holonomic and determines a foliation on M.
Thus, two points can be joined by a trajectory of the system if and only if they are situated on the
same leaf of the foliation.

3 Production management problem
We consider that a company can manufacture n types of products, denoted P1, P2,..., Pn. In a

certain period of time T (fixed) it must produce a certain amount (s1,s2, ...,sn) of each type of product.
The quantity of Pn product depends on the quantities of products P1, P2 ,..., Pn−1 by a given law. We
know that the unit storage costs of holding inventory are given by constants (β1,β2, ...,βn) for each



doi.org/10.15837/ijccc.2020.2.3859 4

product. Also, we assume that the unit production costs increase linearly with the production level
and the cost of production operations for Pn are considered negligible (Pn is a final product packed
and unassembled). We are looking for a plan of production for filling the order at the specified delivery
data at minimum cost. The case of a single product is studied by Kamien and Schwartz [12]. We
consider xi = xi(t) the inventory accumulated by time t. The inventory level is the cumulated past
production pi = pi(t) and considering xi(0) = 0, we have

xi(t) =
∫ t

0
pi(s)ds.

Hence the rate of change of inventory level ẋi is the production and we have ẋi = pi. The unit
production costs ci increase linearly with the production level, i.e ci = αipi, where α1, ...,αn−1 ∈ (0, 1)
are positive constants and we have that the total cost of production is

c1p
1 + ... + cn−1pn−1 =

n−1∑
i=1

αi
(
pi
)2

=
n−1∑
i=1

αi
(
ẋi
)2
.

We obtain that the total cost (including storage), at time t is

n−1∑
i=1

αi
(
ẋi
)2

+
n∑
i=1

βix
i.

Considering, ẋi = ui, i = 1,n−1, the control variables and assuming that the rate of change of
inventory for final product Pn is given by the law

ẋn(t) = u1
x1(t)
k1

+ ... + un−1
xn−1(t)
kn−1

,

where ki > 0, i = 1,n− 1, we obtain the following optimal control problem


ẋ1 = u1
ẋ2 = u2
.....
ẋn−1 = un−1

ẋn = u1 x
1

k1
+ ... + un−1 x

n−1

kn−1

xi(0) = 0, xi(T) = si, i = 1,n
u1, ...,un−1 ≥ 0

(4)

min
∫ T

0

(
α1(u1)2 + ... + αn−1(un−1)2 + β1x1 + ... + βnxn

)
dt.

We are looking for the optimal solutions starting from the point (0, 0, ..., 0) and (s1,s2, ...sn) as end-
point. The system can be written in the form (driftless control affine system):

ẋ =
n−1∑
i=1

uiXi, x =



x1

...
xn


 , X1 =




1
0
...
x1

k1


 , ...,Xn−1 =




0
...
1

xn−1

kn−1


 ∈ Rn+ (5)

min
∫ T

0
F(u(t),x(t))dt,

F(u(t),x(t)) = α1(u1)2 + ... + αn−1(un−1)2 + β1x1 + ... + βnxn.

The distribution ∆ = span{X1, ...,Xn−1} which is generated by the vector fields X1, ...,Xn−1 has
constant dimension, dim ∆(x) = n− 1, for all x ∈ Rn. The vector fields are given by

X1 =
∂

∂x1
+
x1

k1

∂

∂xn
, X2 =

∂

∂x2
+
x2

k2

∂

∂xn
, ...,Xn−1 =

∂

∂xn−1
+
xn−1

kn−1

∂

∂xn
.



doi.org/10.15837/ijccc.2020.2.3859 5

The Lie bracket are

[Xi,Xj] =
[
∂

∂xi
+
xi

ki

∂

∂xn
,
∂

∂xj
+
xj

kj

∂

∂xn

]
= 0, i,j ∈ 1,n− 1

and it results that the associated distribution ∆ = span{X1, ...,Xn−1} is holonomic and has the
constant rank n − 1. Using the Frobenius theorem, it results that the system is not controllable, in
the sense that we cannot reach any final stock quantity. Moreover, from the system (4) we obtain

ẋn =
ẋ1x1

k1
+ ... +

ẋn−1xn−1

kn−1
,

which yields, by integration

xn =
(
x1
)2

2k1
+ ... +

(
xn−1

)2
2kn−1

+ c. (6)

(c is a constant) and it results that ∆ determines a foliation on Rn+ given by the hipersurfaces (6).
From the initial condition xi(0) = 0 it results c = 0 and using that xi(T) = si we have that the system
is controllable (the problem has the solution) if and only if the final amounts satisfy the condition

sn =
s21
2k1

+ ... +
s2n−1
2kn−1

.

To solve the optimal control problem we can use the Pontryagin Maximum Principle on the cotangent
space. We get the Hamiltonian

H(u,x,p) =
n∑
i=1

piẋ
i −F = p1u1 + p2u2 + ... + pn

(
u1
x1

k1
+ ... + un−1

xn−1

kn−1

)
−α1(u1)2 − ...−αn−1(un−1)2 −β1x1 − ...−βnxn,

The conditions ∂H
∂ui

= 0 lead to the following formulas of control variables

ui =
piki + pnxi

2αiki
, (7)

which replaced into the expression of the Hamiltonian function leads to

H =
n−1∑
i=1

(
piki + pnxi

)2
4αik2i

−β1x1 − ...−βnxn. (8)

Using the Hamilton-Jacobi-Bellman equations (1) we obtain a very complicated system of differential
equations. Therefore, we will use a different way, involving the framework of Lie algebroids.

3.1 Lie algebroids

We consider M a real, C∞-differentiable, n-dimensional manifold and TxM its tangent space in
the point x ∈ M. The tangent bundle of M is usually noted (TM,πM,M), where πM is the canonical
projection map πM : TM → M applying a tangent vector X(x) ∈ TxM ⊂ TM to the base point
x ∈ M. A vector bundle is a triple denoted (E,π,M) with E the total space, M the base space and
the map π : E → M is a surjective submersion. Using Mackenzie [17] we have:

Definition 1. A Lie algebroid is given by (E, [·, ·]E,σ), where (E,π,M) is a vector bundle of rank m
over the manifold M, satisfying the following conditions:
i) C∞(M)-module of sections Γ(E) is endowed with a Lie algebra structure [·, ·]E.
ii) The map σ : E → TM (called the anchor) induces a Lie algebra homomorphism from the Lie
algebra of sections (Γ(E), [·, ·]E) to the Lie algebra of vector fields (�(M), [·, ·]) and verifies the Leibniz
rule

[s1,fs2]E = f[s1,s2]E + (σ(s1)f)s2, ∀s1,s2 ∈ Γ(E), f ∈ C∞(M).



doi.org/10.15837/ijccc.2020.2.3859 6

Considering the local coordinates (xi) on an open subset U ⊂ M and a local basis {sα} of the
sections of the bundle π−1(U) → U, these generate the local coordinates (xi,yα) on E. The local
functions σiα(x), L

γ
αβ(x) on the base M given by following relations

σ(sα) = σiα
∂

∂xi
, [sα,sβ]E = L

γ
αβsγ, i = 1,n, α,β,γ = 1,m,

are called the structure functions of Lie algebroids. We recall that a control system on the Lie algebroid
(E, [·, ·]E,σ) (see Martinez [16]) with the control space τ : A → M is given by a section ρ of E along τ.
A trajectory of the control system ρ is an integral curve of the vector field σ(ρ). Considering the cost
function L∈ C∞(A), we must minimize the integral of L over the set of those system trajectories which
satisfy certain optimal conditions. The Hamiltonian function H is given by H(µ,u) = 〈µ,ρ(u)〉−L(u)
and the critical trajectories are given by Hamilton-Jacobi equations on Lie algebroids (Martinez [16])

dxi

dt
= σiα

∂H
∂µα

,
dµα
dt

= −σiα
∂H
∂xi
−µγL

γ
αβ

∂H
∂µβ

. (9)

with ∂H
∂uA

= 0. Using [20] we have:

Remark 3.1. The connection between the Hamiltonian function H on the dual space T∗M and the
Hamiltonian function H on the dual Lie algebroid E∗ has the form

H(p) = H(σ?(p)), µ = σ?(p), p ∈ T∗xM, µ ∈ E
∗
x. (10)

which locally yields

µα = σ∗iα pi, (11)

and the Hamiltonian H(p) is degenerate on the subset Kerσ? ⊂ T∗M.

3.2 Solution of production management problem

We will solve the problem of production management using the framework of Lie algebroids.
Thus, we consider the space E = ∆ (holonomic distribution with constant rank), the anchor map
σ : E → TM is given by inclusion and [, ]E is the induced Lie bracket. The anchor σ has the
components

σiα =




1 0 ... 0
0 1 ... 0
...

... 0
0 0 ... 1
x1

k1
x2

k2
... x

n−1

kn−1


 ,

and we get the Lagrangian function L on E given by

L = α1(u1)2 + ... + αn−1(un−1)2 + β1x1 + ... + βnxn,

which is regular, because det
(

∂2L
∂ui∂uj

)
6= 0. Next, we use the Legendre transformation in order to find

the Hamiltonian function on dual Lie algebroid.

Theorem 2. The Hamiltonian function on dual Lie algebroid E∗ has the following expression

H(x,µ) =
n−1∑
i=1

µ2i
4αi
−β1x1 − ...−βnxn. (12)

Proof. The Legendre transformation from E to E∗ induced by the regular Lagrangian L has the form

(x,u) → (x,µ), µi = Φi(x,u) =
∂L
∂ui

= 2αiui,



doi.org/10.15837/ijccc.2020.2.3859 7

and the Hamiltonian function is given by H(x,µ) = µΦ−1(x,µ)−L(x, Φ−1(x,µ)). In (x,u) coordinates
we get

H = ui
∂L
∂ui
−L = α1(u1)2 + ... + αn−1(un−1)2 −β1x1 − ...−βnxn,

and using the relations ui = µi2αi , we find (12). ut

Using the relation (11) we can find the Hamiltonian function H on T∗M in the form H(x,p) =
H(µ), µ = σ?(p), with




µ1
...

µn−1


 =




1 0 ... 0 x
1

k1

0 1 ... 0 x
2

k2
...

...
...

...
0 0 ... 1 x

n−1

kn−1






p1
...

pn−1
pn


 .

We get that

µi =
piki + pnxi

ki

and it results the Hamiltonian on the dual space, given in relation (8).

Theorem 3. The optimal solution of production management problem is given by

xi(t) = ai1e
√

βn
2αiki

t
+ ai2e

−
√

βn
2αiki

t
−
βiki
βn

, i = 1,n− 1, (13)

xn(t) =
(
x1(t)

)2
2k1

+ ... +
(
xn−1(t)

)2
2kn−1

, (14)

where

ai1 =
1

di + 1
βiki
βn

+
disi
d2i − 1

, ai2 =
di

di + 1
βiki
βn
−

disi
d2i − 1

, di = e
√

βn
2αiki

T
, (15)

and control variables

ui(t) =
√

βn
2αiki


ai1e

√
βn

2αiki
t
−ai2e

−
√

βn
2αiki

t


 . (16)

Proof. From the relation [Xα,Xβ] = L
γ
αβXγ we obtain the components L

γ
αβ = 0, while from (9) we

deduce that
dxi

dt
= σiα

∂H
∂µα

,
dµα
dt

= −σiα
∂H
∂xi

,

which leads to 


ẋi = µi2αi , i = 1,n− 1
ẋn =

∑n−1
i=1

µix
i

2αiki
,

µ̇i = βi + x
i

ki
βn, i = 1,n− 1

(17)

We deduce, using (17)
ẍi =

µ̇i
2αi
⇒ ẍi =

βn
2αiki

xi +
βi

2αi
, i = 1,n−1,

which yields the linear nonhomogeneous second order differential equations. Next, considering the
linear homogeneous differential equations

ẍi −
βn

2αiki
xi = 0,



doi.org/10.15837/ijccc.2020.2.3859 8

and the characteristic equation λ2 − βn2αiki = 0, we get the solutions λ1,2 = ±
√

βn
2αiki

. It results the
general solutions of the homogeneous differential equations

xi(t) = ai1e
√

βn
2αiki

t
+ ai2e

−
√

βn
2αiki

t
.

Also, we find the general solution of the nonhomogeneous second order differential equations given by
(13). The solution is optimal because the Hamiltonian is a convex function. The constants ai1,ai2
can be found from the initial conditions xi(0) = 0, xi(T) = si, which lead to the system{

ai1 + ai2 =
βiki
βn

ai1di +
1
di
ai2 =

βiki
βn

+ si

where we denote di = e
√

βn
2αiki

T
> 1. From the above system, by direct computation, the coefficients

(15) are obtained. Finally, using (4) we have ẋi = ui, i = 1,n− 1 and it results the control variables
(16), which ends the proof. ut

Numerical application: We consider a numerical example for the case n = 3:
- fixed period of time is T = 1;
- final stock quantities from products P1, P2, P3 are s1 = 6, s2 = 4, s3 = 8;
- storage costs are given by β1 = 4, β2 = 2, β3 = 4;
- the coefficients are α1 = 2/3, α2 = 1/2, k1 = 3, k2 = 4.
The economic system is controllable, because

s3 =
s21
2k1

+
s22
2k2

= 8.

If s3 6= 8, then the system is not controllable. Applying the formulas (13)-(16) the following optimal
solution is obtained

x1(t) = 3, 3595786488et − 0, 3595786488e−t − 3,
x2(t) = 2, 2397190992et − 0, 2397190992e−t − 2,

x3(t) =
(
x1(t)

)2
6

+
(
x2(t)

)2
8

,

with control variables (production rates) given by

u1(t) = 3, 3595786488et + 0, 3595786488e−t,
u2(t) = 2, 2397190992et + 0, 2397190992e−t.

4 Conclusions and future work
In this paper a mathematical model for a problem of production management is proposed. We have

to find the optimal solution so that different products are manufactured in the required quantities and
at a fixed date with minimum production and storage costs. The continuous optimal control problem
is solved by using the Pontryagin Maximum Principle at the level of a new working space, called Lie
algebroid, which is a holonomic distribution on tangent spaces. The controllability of our economic
system is solved by using the Lie geometric methods and the properties of the Lie brackets for the
vectors fields which generate the distribution. The system is controllable if and only if the final stock
quantities satisfy a given law. Finally, a numerical application is presented. We have to remark that
the optimal solution of this numerical example can be easily represented in three-dimensional space
using mathematical software.
As future work, we will consider the case of nonholomic and strong bracket generating distribution
[20], where the system is controllable. This type of distributional system has many applications
in automatic control, see for instance the paper [6], where the minimum wheel-rotation paths for
differential-drive mobile robots are studied and the mathematical model is a driftless control affine
system. It is interesting to use the PMP on the Lie algebroid, which in this case is the whole tangent
space with the base given by vectors of distribution, together with the first iterated Lie brackets.



doi.org/10.15837/ijccc.2020.2.3859 9

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Cite this paper as:
Popescu, L.; Militaru, N.D.; Mituca, O.M.(2020). Optimal Control Applications in the Study

of Production Management, International Journal of Computers Communications & Control, 15(2),
3859, 2020.

https://doi.org/10.15837/ijccc.2020.2.3859