Optimal control applications in the study of inventory and production problems [615239]

Mathematical Methods of Operations Research
Optimal control applications in the study of inventory and production problems
–Manuscript Draft–

Manuscript Number: MMOR-D-19-00005
Full Title: Optimal control applications in the study of inventory and production problems
Article Type: Original Research
Section/Category: nonlinear optimization and game theory
Keywords: inventory and production,
optimal control,
controllability,
Hamilton-Jacobi-Bellman equations,
Lie algebra
Corresponding Author: Liviu Popescu, Ph.D
Universitatea din Craiova
Craiova, Dolj ROMANIA
Corresponding Author Secondary
Information:
Corresponding Author's Institution: Universitatea din Craiova
Corresponding Author's Secondary
Institution:
First Author: Liviu Popescu, Ph.D
First Author Secondary Information:
Order of Authors: Liviu Popescu, Ph.D
Order of Authors Secondary Information:
Funding Information:
Abstract: We solve a problem of inventory and production using the optimal control techniques
and the Pontryagin Maximum Principle at the level of a new working space called Lie
algebroid. We prove that the framework of a Lie algebroid is more suitable than the
cotangent space in order to find the optimal solutions of a driftless control affine system
with integrable distribution.
Suggested Reviewers: Vasile Georgescu, Ph.D
Professor, Universitatea din Craiova
[anonimizat]
He study the nonlinear optimization.
Jose Carinena, Ph.D
Professor, Universidad de Zaragoza
[anonimizat]
He study applications of Lie algebroids.
Yuri Sachkov, Ph.D
Professor, Program Systems Institute
[anonimizat]
He study control theory.
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Optimal control applications in the study of inventory and production
problems
Liviu Popescu
Abstract
We solve a problem of inventory and production using the optimal control techniques and the
Pontryagin Maximum Principle at the level of a new working space called Lie algebroid. We prove
that the framework of a Lie algebroid is more suitable than the cotangent space in order to find the
optimal solutions of a driftless control affine system with integrable distribution.
JEL Classification: C61, C65, C44
Keywords : inventory and production, optimal control, controllability, Hamilton-Jacobi-Bellman
equations, Lie algebra.
1 Introduction
It is well known that the mathematical methods have been applied successfully in different domains of
research as optimal control theory with applications in economics, business administration, finance or
engineering. One of the motivations for this work is the study of Lagrangian systems with some external
holonomic constraints. These systems have a wide application in many different areas as optimal control
theory, econometrics, cybernetics or operational research (see Arrow (1968), Anita, Arn˘ autu, & Capasso
(2011), Caputo (2005), Feichtinger, Hartl, & Kort (2001), Seierstad & Sydsater (1987), Sethi & Thompson
(2000), Weber, (2011)).
Also, Lie geometric methods in the control theory have been applied by many authors, see for instance
Agrachev & Sachkov (2004), Brocket (1973), Isidori (1995), Jurdjevic (1997), LaValle (2006), Popescu
(2005), (2017) and references therein. One of the most important methods in the geometric approach
is the analysis of the solution for the optimal control problem as provided by Pontryagin’s Maximum
Principle. A curve c(t) = ( x(t); u(t)) is an optimal trajectory if there exists a lifting of x(t) to the
dual space ( x(t); p(t)) satisfying the Hamilton-Jacobi-Bellman equations. However, finding a complete
solution to an optimal control problem remains extremely difficult for several reasons. First of all, we are
dealing with the problem of integrating a Hamiltonian system, which is generally difficult to integrate,
except for particular dynamics and costs. Secondly, some special solutions so-called abnormal, should be
studied. Finally, even if all solutions are found, there remains the problem of selecting optimal solutions
from them. For these reasons, it is important to find new methods and new working spaces that simplify
the study.
1
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In this paper we propose a mathematical model for a problem of inventory and production, using the
optimal control techniques. We find the optimal solution applying the Pontrygin Maximum Principle
on a Lie algebroid, which in this case is a holonomic distribution of the tangent space. Such problems
of inventory and production are intensely studied (see for example AL-Khazraji, Cole & Guo (2017),
Chazal, Jouini & Tahraoui (2008), Gayon, Vercraene & Flapper (2017), Karaman (2017), Maccini, Moore
& Schaller (2015), Olsson (2018), Ortega & Lin (2004). Moreover, we show that the framework of Lie
algebroids is more suitable than the cotangent space in the study of driftless control affine systems with
holonomic distributions. The controllability of economical system is studied using Lie geometric methods
and the Frobenius theorem.
The 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 controllability
problems in the case of holonomic distributions. In section three, we give an application of driftless
control affine system to a problem of inventory and production and show that the Hamilton-Jacobi-
Bellman equations, provided by Pontryagin Maximum Principle on cotangent space, lead to a very
complicated system of differential equations. We will use a different approach considering the framework
of Lie algebroids. We present only the necessary notions about Lie algebroids (see Mackenzie (1987)
for more details) and the geometric viewpoint of the optimal control. We find the relation between the
Hamiltonian Hon dual Lie algebroid and the Hamiltonian Hon the cotangent space, that is very useful
in the study of control affine systems. The strategy is to apply the Pontryagin Maximum Principle at
the level of Lie algebroids. Moreover, we prove that the distribution generated by the vector fields is
holonomic and it determines a foliation in the state space. In the last part of the paper we find the
complete solution of the problem using the Pontryagin Maximum Principle and the framework of a Lie
algebroid.
2 Theoretical Basic
2.1 Control affine systems
LetMbe a smooth n-dimensional manifold. We consider the control system given by differential
equations, depending on some parameters
dxi
dt=fi(x; u);
where x2Mrepresents the state of the system and u2U½Rmrepresents the controls . Let x0andx1
be two points of M. An optimal control problem consists of finding the trajectories of our control system
which connect x0andx1and minimizing the cost
minZT
0L(x(t); u(t))dt; x (0) = x0; x(T) =x1;
where Lis the Lagrangian orrunning cost (energy, cost, time, distance, etc.). Control theory deals with
systems whose evolution can be influenced by some external agents. The most important and powerful
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tool for studying the optimal solutions in control theory is Pontryagin’s Maximum Principle. It generates
the differential 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 is given by
H(x; p; u ) =hp; f(x; u)i ¡L(x; u); p2T¤M;
while the maximization condition with respect to the control variables u, namely
H(x(t); p(t); u(t)) = maxvH(x(t); p(t); v);
leads to@H
@u= 0 ( His assumed to be smooth with respect to u) and the extreme trajectories satisfy the
equations
˙x=@H
@p;˙p=¡@H
@x: (1)
Definition 1 A control affine system has the form, see LaValle (2006)
˙x=X0(x) +mX
i=1uiXi(x); (2)
where x= (x1; :::; x n)are local coordinates on a smooth ndimensional manifold M,u(t) = (u1(t); :::; u m(t))
2U½Rm,m·nandX0; X1:::Xmare smooth vector fields on M.
Usually, X0is called the drift vector field describing the dynamics of the system in the absence of
controls, and the vector fields Xi; i=1; mare called the input vector fields. The function u(t) is called
the control or the input function, which may be specified freely in order to steer the system in a desired
direction.
Definition 2 The system is controllable if for any two points x0andx1onMthere exists a finite T
and an admissible control u: [0; T]!Usuch that for xsatisfying x(0) = x0we have that x(T) =x1.
In the other words, the system is controllable if for any two states x0,x1, there exists a solution
curve of (2) connecting x0tox1. Controllability is the ability to steer a system from a given initial state
to any final state, in finite time, using the available controls. The reachable set Rof a point x02M
characterizes the states x2Mthat can be reached from a given initial state x0in positive time, by
choosing various controls and switching from one to another from time to time. A system is controllable
ifR(x) =M;8x2M. Controllability doesn’t care about the quality of the trajectory between two
states, neither for the amount of control effort.
Definition 3 A distribution ∆on the manifold Mis a map which assigns to each point in Ma subspace
of the tangent space at this point
M3x!∆(x)½TxM:
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The distribution ∆ is called locally finitely generated if there is a family of vector fields fXigi=1;m
(called local generators of ∆) which spans ∆, i.e. ∆( x) =spanfX1(x); :::; X m(x)g ½TxM. The distri-
bution ∆ has dimension kif dim ∆( x) =k, for all points xinM. We recall that the Lie bracket of two
vector fields is given by
[f; g](x) =@g
@x(x)f(x)¡@f
@x(x)g(x);
(@g
@xis the Jacobian matrix of g). A distribution ∆ on Mis said to be involutive if, 8x2Mthen
f(x); g(x)2∆(x))[f; g] (x)2∆(x):
If the involutive distribution is generated by vector fields fXigi=1;mthen we have
[Xi; Xj] (x) =mX
k=1Lk
ij(x)Xk(x):
In other words, every Lie bracket can be expressed as a linear combination of the system vector fields, and
therefore it already belongs to ∆. The Lie brackets are unable to escape ∆ and generate new directions
of motion. We recall that a foliation fS®g®2AofMis a partition of M=[
®2AS®ofMinto disjoint
connected submanifolds S®, called leaves.
Definition 4 A distribution ∆of constant dimension on Mis called integrable (holonomic) if there
exists a foliation fS®g®2AonMwhose tangent bundle is ∆, that is TxS= ∆( x), where Sis the leaf
passing through x.
Theorem 5 (Frobenius) Supposing that a distribution ∆has constant dimension, then ∆is integrable if
and only if ∆is involutive.
The presence of the drift X0in the study of control affine systems, significantly complicates the
question of controllability. In the following we consider the driftless control affine system (X0= 0) in
the form
˙x=mX
i=1uiXi(x): (3)
The vector fields Xi; i=1; m, generate a distribution ∆ on M(assumed to be connected) so that the
rank of ∆ is constant. We will characterize the controllability using the properties of vector fields which
generate the distribution ∆. If the distribution ∆ = spanfX1; X2; :::; X mgis holonomic with constant
rank, then [ Xi; Xj]2Dfor every i; j=1; m,i6=jand the system is not controllable. From the Frobenius
theorem, the distribution ∆ is integrable, it determines a foliation on Mand two points can be joined if
and only if they are situated on the same leaf.
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3 Inventory and production problem
We consider that a company manufactures ntypes of products, denoted P1,P2,…,Pn. In a certain period
of time T(fixed) it must to produce a certain amount ( s1; s2; :::; s n) of each type of product. Some of the
quantities of products P1,P2,…,Pn¡1are used in the manufacture of the Pnproduct. We know the unit
storage costs of holding inventory ( ¯1; ¯2; :::; ¯ n) for each product. Also, the unit production costs rise
linearly with the production level and the cost of production operations for Pnare considered negligible.
We are looking for a plane 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 (2006). Let us 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) =Zt
0pi(s)ds:
Hence the rate of change of inventory level ˙ xiis the production and we have ˙ xi=pi. The unit production
costs cirise linearly with the production level, i.e ci=®ipi, where ®1; :::; ® n¡1are positive constants and
we have that the total cost of production is
c1p1+:::+cn¡1pn¡1=n¡1X
i=1®i³
pi´2=n¡1X
i=1®i³
˙xi´2:
We obtain that the total cost (including storage), at time tis
n¡1X
i=1®i³
˙xi´2+nX
i=1¯ixi:
Considering, ˙ xi=ui,i=1; n¡1;the control variables and assuming that the rate of change of inventory
forPnis given by the law
˙xn=u1x1
k1+:::+un¡1xn¡1
kn¡1;
where ki>0,i=1; n¡1, we obtain the following optimal control problem
8
>>>>>>>>>><
>>>>>>>>>>:˙x1=u1
˙x2=u2
:::::
˙xn¡1=un¡1
˙xn=u1x1
k1+:::+un¡1xn¡1
kn¡1
xi(0) = 0 ; xi(T) =si; i=1; n
u1; :::; un¡1¸0(4)
min
u(¢)ZT
0³
®1(u1)2+:::+®n¡1(un¡1)2+¯1×1+:::+¯nxn´
dt;
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We are looking for the optimal solutions starting from the point (0 ;0; :::;0) and ( s1; s2; :::s n) as endpoint.
The system can be written in the form (driftless control affine system):
˙x=n¡1P
i=1uiXi; x =0
B@x1
…
xn1
CA2Rn
+; X1=0
BBBB@1
0
…
x1
k11
CCCCA; X2=0
BBBB@0
1
…
x2
k21
CCCCA; :::; X n¡1=0
BBBB@0
…
1
xn¡1
kn¡11
CCCCA2Rn
+
min
u(¢)RT
0F(u(t); x(t))dt;F(u(t); x(t)) =®1(u1)2+:::+®n¡1(un¡1)2+¯1×1+:::+¯nxn:(5)
The distribution ∆ = spanfX1; :::; X n¡1gwhich is generated by the vector fields X1; :::; X n¡1has constant
dimension, dim ∆( x) =n¡1, for all x2Rn. The vector fields are given by
X1=@
@x1+x1
k1@
@xn; X 2=@
@x2+x2
k2@
@xn; :::; X n¡1=@
@xn¡1+xn¡1
kn¡1@
@xn:
The Lie bracket are
[Xi; Xj] ="
@
@xi+xi
ki@
@xn;@
@xj+xj
kj@
@xn#
= 0; i; j 21; n¡1
and it results that the associated distribution ∆ = spanfX1; :::; X n¡1gis holonomic and has the constant
rank n¡1. 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
˙xn=˙x1x1
k1+:::+˙xn¡1xn¡1
kn¡1;
which yields
xn=¡x1¢2
2k1+:::+¡xn¡1¢2
2kn¡1+c: (6)
(cis 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) =siwe have that the system is
controllable (the problem has the solution) if and only if the final amounts satisfy the condition
sn=s2
1
2k1+:::+s2
n¡1
2kn¡1:
In order to solve this optimal control problem we can use the Pontryagin Maximum Principle on the
cotangent space. We get the Hamiltonian
H(u; x; p ) =n¡1X
i=1pi˙xi¡ F=p1u1+p2u2+:::+pnÃ
u1x1
k1+:::+un¡1xn¡1
kn¡1!
¡®1(u1)2¡:::¡®n¡1(un¡1)2¡¯1×1¡:::¡¯nxn;
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The conditions@H
@ui= 0 lead to the following formulas of control variables
ui=piki+pnxi
2®iki; (7)
which replaced into the expression of the Hamiltonian function leads to
H=n¡1X
i=1¡piki+pnxi¢2
4®ik2
i¡¯1×1¡:::¡¯nxn: (8)
Using the Hamilton-Jacobi-Bellman equations (1) we obtain the following system
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>>>>>>><
>>>>>>>:˙xi=@H
@pi=piki+pnxi
2®iki; i=1; n¡1
˙xn=@H
@pn=n¡1P
i=1(piki+pnxi)xi
2®ik2
i;
˙pi=¡@H
@xi=¯i¡(piki+pnxi)pn
2®ik2
i; i=1; n¡1
˙pn=¡@H
@xn=¯n:
which is a very complicated system of differential equations. For this reason we will use a different
approach, involving the framework of Lie algebroids.
4 Lie algebroids
LetMbe a real, C1-differentiable, n-dimensional manifold and TxMits tangent space at x2M. The
tangent bundle of Mis denoted ( TM; ¼ M; M);where TM=[
x2MTxMand¼Mis the canonical projection
map ¼M:TM!Mtaking a tangent vector X(x)2TxM½TM to the base point x2M:A vector
bundle is a triple ( E; ¼; M ) where EandMare manifolds, called the total space and the base space, and
the map ¼:E!Mis a surjective submersion. Using Mackenzie (1987) we have:
Definition 6 A Lie algebroid over a manifold Mis a triple (E;[¢;¢]E; ¾), where (E; ¼; M )is a vector
bundle of rank mover M;which satisfies the conditions:
a)C1(M)-module of sections Γ(E)is equipped with a Lie algebra structure [¢;¢]E.
b)¾:E!TM is a bundle map, called the anchor, which induces a Lie algebra homomorphism (also
denoted ¾) from the Lie algebra of sections (Γ(E);[¢;¢]E)to the Lie algebra of vector fields (Â(M);[¢;¢])
satisfying the Leibniz rule
[s1; fs2]E=f[s1; s2]E+ (¾(s1)f)s2;8s1; s22Γ(E); f2C1(M):
Also, it results:
1±[¢;¢]Eis aR-bilinear operation,
2±[¢;¢]Eis skew-symmetric, i.e.
[s1; s2]E=¡[s2; s1]E;8s1; s22Γ(E);
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3±[¢;¢]Everifies the Jacobi identity
[s1;[s2; s3]E]E+ [s2;[s3; s1]E]E+ [s3;[s1; s2]E]E= 0;
and¾being a Lie algebra homomorphism, we have
¾[s1; s2]E= [¾(s1); ¾(s2)]:
Iffis a function on M, then d f(x)2E¤
xis given by hd f(x); ai=¾(a)f, for8a2Ex. For !2Vk(E¤)
theexterior derivative dE!2Vk+1(E¤) is given by the formula
dE!(s1; :::; s k+1) =k+1X
i=1(¡1)i+1¾(si)!(s1; :::;ˆsi; :::; s k+1) +
+X
1·i<j·k+1(¡1)i+j!([si;sj]E; s1; :::;ˆsi; :::;ˆsj; :::s k+1):
where si2Γ(E),i=1; k+ 1, and the hat over an argument means the absence of the argument. It
results that ( dE)2= 0. If we take the local coordinates ( xi) on an open U½M, a local basis fs®gof the
sections of the bundle ¼¡1(U)!Ugenerates local coordinates ( xi; y®) onE. The local functions ¾i
®(x),
L°
®¯(x) onMgiven by
¾(s®) =¾i
®@
@xi;[s®; s¯]E=L°
®¯s°; i=1; n; ®; ¯; ° =1; m;
are called the structure functions of Lie algebroids. Some examples of Lie algebroids are:
Example 7 The tangent bundle E=TM itself, with identity mapping as anchor. With respect to the
usual coordinates (x;¢x), the structure functions are Li
jk= 0,¾i
j=±i
j, but if we were to change to another
basis for the vector fields, the structure functions would become nonzero.
Example 8 Any integrable subbundle of TM is a Lie algebroid with the inclusion as anchor and the
induced Lie bracket.
By a control system on the Lie algebroid ( E;[¢;¢]E; ¾) (see Martinez (2004)) with the control space
¿:A!Mwe mean a section ½ofEalong ¿. A trajectory of the system ½is an integral curve of the
vector field ¾(½). Given the cost function L 2C1(A), we have to minimize the integral of Lover the
set of those system trajectories which satisfy certain boundary conditions. The Hamiltonian function
H 2C1(E¤£MA) is defined by
H(¹; u) =h¹; ½(u)i ¡ L (u);
whereas the associated Hamiltonian control system ½His given by the symplectic equation on Lie algebroid
i½H!E=dEH:
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where !Eis the canonical symplectic form. The critical trajectories are given by Hamilton-Jacobi-Bellman
equations on Lie algebroids, Martinez (2004)
@H
@uA= 0;dxi
dt=¾i
®@H
@¹®;d¹®
dt=¡¾i
®@H
@xi¡¹°L°
®¯@H
@¹¯: (9)
Theorem 9 The relation between the Hamiltonian function Hon the cotangent bundle T¤Mand the
Hamiltonian function Hon the dual Lie algebroid E¤is given by
H(p) =H(¾?(p)); ¹ =¾?(p); p2T¤
xM; ¹ 2E¤
x: (10)
Proof. We can associate to any Lagrangian function L:E!Ron Lie algebroids E;a Lagrangian
LonIm¾½TMdefined by
L(v) =fL(u)ju2Ex; ¾(u) =vg;
where v2(Im¾)x½TxM,x2M. The Fenchel-Legendre dual of Lagrangian Lis the Hamiltonian H
given by
H(p) = sup
vfhp; vi ¡L(v)g= sup
vfhp; vi ¡ L (u);¾(u) =vg
= sup
ufhp; ¾(u)i ¡ L (u)g= sup
ufh¾?(p); ui ¡ L (u)g=H(¾?(p));
and we get
H(p) =H(¹); ¹ =¾?(p); p2T¤
xM; ¹ 2E¤
x;
which locally leads to
¹®=¾¤i
®pi; (11)
where the Hamiltonian H(p) is degenerate on Ker¾?½T¤M: u t
4.1 Solution of the inventory and production problem
In order to solve our problem of inventory and production we use the framework of Lie algebroids and
consider E= ∆ (holonomic distribution with constant rank), the anchor ¾:E!TM is the inclusion
and [ ;]Ethe induced Lie bracket. The anchor ¾has the components
¾i
®=0
BBBBBB@1 0 ::: 0
0 1 ::: 0
…… 0
0 0 ::: 1
x1
k1x2
k2:::xn¡1
kn¡11
CCCCCCA;
and we get te Lagrangian function LonEgiven by
L=®1(u1)2+:::+®n¡1(un¡1)2+¯1×1+:::+¯nxn;
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which is regular, because det³
@2L
@ui@uj´
6= 0, where
@2L
@ui@uj=0
BBBB@2®10::: 0
0 2 ®2::: 0
………
0 0 :::2®n¡11
CCCCA
Next, we use the Legendre transformation in order to find the Hamiltonian function on dual Lie algebroid.
Proposition 10 The Hamiltonian function on dual Lie algebroid E¤is given by
H(x; ¹) =n¡1X
i=1¹2
i
4®i¡¯1×1¡:::¡¯nxn: (12)
Proof . The Legendre transformation from EtoE¤induced by the regular Lagrangian Lhas the
form
(x; u)!(x; ¹); ¹ i= Φ i(x; u) =@L
@ui= 2®iui;
and the Hamiltonian function is given by
H(x; ¹) =¹Φ¡1(x; ¹)¡ L(x;Φ¡1(x; ¹)):
In (x; u) coordinates we get
H=ui@L
@ui¡ L=®1(u1)2+:::+®n¡1(un¡1)2¡¯1×1¡:::¡¯nxn;
and using the relations
ui=¹i
2®i
we find (12). u t
Using (11) we can calculate the Hamiltonian HonT¤Min the form H(x; p) =H(¹); ¹=¾?(p),
where
0
B@¹1
…
¹n¡11
CA=0
BBBBB@1 0 :::0x1
k1
0 1 :::0x2
k2…………
0 0 :::1xn¡1
kn¡11
CCCCCA0
BBBB@p1
…
pn¡1
pn1
CCCCA:
We get that
¹i=piki+pnxi
ki
and it results the Hamiltonian on the cotangent bundle, given in (8).
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Theorem 11 The optimal solution of inventory and production problem is given by
xi(t) =ai1eq
¯n
2®ikit+ai2e¡q
¯n
2®ikit¡¯iki
¯n; i=1; n¡1;
xn(t) =¡x1¢2(t)
2k1+:::+¡xn¡1¢2(t)
2kn¡1;
where
ai1=1
di+ 1¯iki
¯n+disi
d2
i¡1; a i2=di
di+ 1¯iki
¯n¡disi
d2
i¡1; d i=eq
¯n
2®ikiT;
and control variables
ui(t) =s
¯n
2®iki0
@ai1eq
¯n
2®ikit¡ai2e¡q
¯n
2®ikit1
A+1
T0
@si¡ai10
@eq
¯n
2®ikiT¡11
A¡ai20
@e¡q
¯n
2®ikiT¡11
A1
A;
with the conditionsµ¯iki
¯n¶2
·4ai1ai2; i=1; n¡1:
Proof . From the relation [ X®; X¯] =L°
®¯X°we obtain the components L°
®¯= 0;while from (9) we
deduce that
dxi
dt=¾i
®@H
@¹®;d¹®
dt=¡¾i
®@H
@xi;
which lead to
8
>>><
>>>:˙xi=¹i
2®i; i=1; n¡1
˙xn=n¡1P
i=1¹ixi
2®iki;
˙¹i=¯i+xi
ki¯n; i=1; n¡1(13)
We deduce
¨xi=˙¹i
2®i)¨xi=¯n
2®ikixi+¯i
2®i; i=1; n¡1;
which yields the linear nonhomogeneous second order differential equations. Next, considering the linear
homogeneous differential equations
¨xi¡¯n
2®ikixi= 0;
from the characteristic equation ¸2¡¯n
2®iki= 0, we get the solutions ¸1;2=§q
¯n
2®iki. It results the
general solutions of the homogeneous differential equations
xi(t) =ai1eq
¯n
2®ikit+ai2e¡q
¯n
2®ikit:
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Also, we find the general solution of the nonhomogeneous second order differential equations given by
xi(t) =ai1eq
¯n
2®ikit+ai2e¡q
¯n
2®ikit¡¯iki
¯n: (14)
The solution is optimal because the Hamiltonian is a convex function.The constants ai1; ai2can be found
from the initial conditions xi(0) = 0, xi(T) =si, which lead to the system
(
ai1+ai2=¯iki
¯n
ai1di+1
diai2=¯iki
¯n+si
where we denote di=eq
¯n
2®ikiT>1. We obtain
ai1=1
di+ 1¯iki
¯n+disi
d2
i¡1; a i2=di
di+ 1¯iki
¯n¡disi
d2
i¡1:
Moreover, the economic conditions xi(t)¸0, for all tgive the following conditions on initial data
µ¯iki
¯n¶2
·4ai1ai2; i=1; n¡1: (15)
Next, using (13) we have
˙¹i=¯i+xi
ki¯n=¯nai1
kieq
¯n
2®ikit+¯nai2
kie¡q
¯n
2®ikit;
which leads to
¹i(t) =s
2®i¯n
ki0
@ai1eq
¯n
2®ikit¡ai2e¡q
¯n
2®ikit1
A+ci:
Hence, we can find the control variables in the form
ui(t) =¹i(t)
2®i=s
¯n
2®iki0
@ai1eq
¯n
2®ikit¡ai2e¡q
¯n
2®ikit1
A+ci
2®i
But, from the relation
xi(T)¡xi(0) =ZT
0˙xi(t)dt=ZT
0ui(t)dt; i =1; n¡1
we get
si=ZT
00
@s
¯n
2®iki0
@ai1eq
¯n
2®ikit¡ai2e¡q
¯n
2®ikit1
A+ci
2®i1
Adt;
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which yields
si=ai10
@eq
¯n
2®ikiT¡11
A+ai20
@e¡q
¯n
2®ikiT¡11
A+ci
2®iT;
and we obtain
ci=2®i
T0
@si¡ai10
@eq
¯n
2®ikiT¡11
A¡ai20
@e¡q
¯n
2®ikiT¡11
A1
A:
Finally, it results the control variables
ui=s
¯n
2®iki0
@ai1eq
¯n
2®ikit¡ai2e¡q
¯n
2®ikit1
A+1
T0
@si¡ai10
@eq
¯n
2®ikiT¡11
A¡ai20
@e¡q
¯n
2®ikiT¡11
A1
A:
u t
Conclusions . In this paper we study a problem of inventory and production using the optimal
control techniques and Lie geometric methods. Besides the economic applications, the novelty of this
study consists in the application of the Pontryagin Maximum Principle at the level of a new working
space, called Lie algebroid, which is in this case an integrable distribution of the tangent space. Also, the
controllability of our economical system is solved using the properties of the Lie brackets for the vector
fields of distribution and involves restrictions on the final stock. Moreover, the economic restrictions of
our problem imply certain conditions on the initial data.
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Author’s address:
University of Craiova,
Dept. of Statistics and Economic Informatics
13, Al. I. Cuza, st. Craiova 200585, Romania
e-mail: liviupopescu@central.ucv.ro; liviunew@yahoo.com
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