An optimal control problem in the study of driftless a ne systems [615238]

An optimal control problem in the study of driftless a¢ ne systems
with holonomic distribution
Aurelia Florea
February 7, 2019
Abstract
In the present paper we study the optimal control problems using Lie geometric methods. We
prove that the framework of Lie algebroids is more suitable than the cotangent bundle in order to
…nd the optimal solutions of driftless control a¢ ne systems with holonomic distribution and positive
homogeneous cost. Finally, we give an application of driftless control a¢ ne system.
MSC2010: 49J15, 93B05, 93B27, 70H05, 17B66
Keywords: control a¢ ne systems, controllability, dynamical systems, optimal control, Hamilton-
Jacobi-Bellman equations, Lie geometric methods.
1 Introduction
In recent years signi…cant applications of dynamical systems and control theory have been witnessed in
diversed areas such as …nance sciences, social sciences, engineering, management and physics. There are
many prominent areas of systems and control theory that include systems governed by linear and nonlinear
ordinary di¤erential equations. Also the Lie geometric methods have been applied successfully in di¤erent
domains of research using dynamical systems or optimal control theory. One of the motivations for this
work is the study of Lagrangian systems with some external holonomic or nonholonomic constraints.
These systems have a wide application in many di¤erent areas as optimal control theory, econometrics,
cybernetics or mathematical economics [2, 3, 5, 7, 8, 16, 17, 19, 20]. The Lie geometric methods in
the control theory have been applied by many authors (see for instance [1, 4, 6, 10, 11, 12, 15, 18] and
references therein). One of the most important method 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, …nding a complete solution to an optimal
control problem remains extremely di¢ cult for several reasons. First of all, we are dealing with the
problem of integrating a Hamiltonian system, which is generally di¢ cult to integrate, except for particular
costs. Secondly, some special solutions so-called abnormal, should be studied. Finally, even if all solutions
1

are found, there remains the problem of selecting optimal solutions from them. For these reasons, it is
important to …nd new methods that simplify the study.
In this paper we prove that the framework of Lie algebroids is more suitable than the cotangent bundle
in the study of driftless control a¢ ne systems with holonomic distribution and positive homogeneous
costs. In the second section we present the known results about Lie geometric methods in optimal
control theory for control a¢ ne systems, including the controllability problems in the case of holonomic
and nonholonomic distributions. Also, we present only the necessary notions about Lie algebroids and the
geometric viewpoint of the optimal control. Also, we …nd the relation between the Hamiltonian Hon dual
Lie algebroid Eand the Hamiltonian Hon the cotangent bundle TM, that is very useful in the study
of control a¢ ne systems. The strategy is to apply the Pontryagin Maximum Principle at the level of Lie
algebroids. Finally, we give an application of driftless control a¢ ne system with positive homogeneous
cost, which is more general than the quadratic cost and show that the Hamilton-Jacobi-Bellman equations,
provided by Pontryagin Maximum Principle on cotangent bundles, lead to a very complicated system of
di¤erential equations. Moreover, it is very di¢ cult to …nd the Hamiltonian function without dependence
on control variables. For these reasons we will use a di¤erent approach considering the framework of
Lie algebroids. However, we prove that the distribution generated by vector …elds is holonomic and it
determines a foliation in three dimensional space. In the last part of the paper we …nd the complete
solution of the problem using the framework of Lie algebroids.
2 Methods used in Optimal Control
LetMbe a smooth n-dimensional manifold. We consider the control system given by di¤erential
equations, depending on some parameters
dxi
dt=fi(x;u);
wherex2Mrepresents the state of the system and u2URmrepresents the controls . Letx0andx1
be two points of M. An optimal control problem consists of …nding the trajectories of our control system
which connects x0andx1and minimizing the cost
minZT
0L(x(t);u(t))dt; x (0) =x0; x(T) =x1;
whereLis the Lagrangian orrunning cost (energy, cost, time, distance, etc.). Control theory deals with
systems whose evolution can be in‡ uenced by some external agents. The most important and powerful
tool for studying the optimal solutions in control theory is Pontryagin’ s Maximum Principle. It generates
the di¤erential equations of …rst order, necessary for the optimal solutions. For each optimal trajectory,
c(t) = (x(t);u(t)), it o¤ers 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)iL(x;u); p2TM;
2

while the maximization condition with respect to the control variables u, namely
H(x(t);p(t);u(t)) = max
vH(x(t);p(t);v);
leads to@H
@u= 0and the extreme trajectories satisfy the equations
_x=@H
@p;_p=@H
@x: (1)
2.1 Controllability of A¢ ne Systems
A control a¢ ne system has the form [12]
_x=X0(x) +mX
i=1uiXi(x); (2)
wherex= (x1;:::;xn)are local coordinates on a smooth ndimensional manifold M,u(t) = (u1(t);:::;um(t))
2URm,mnandX0;X1:::Xmare smooth vector …elds on M. Usually,X0is called the drift vector
…eld describing the dynamics of the system in the absence of controls, and the vector …elds Xi; i=1;m
are called the input vector …elds. The function u(t)is called the control or the input function, which may
be speci…ed freely in order to steer the system in a desired direction.
De…nition 1 The system is controllable if for any two points x0andx1onMthere exists a …nite T
and an admissible control u: [0;T]!Usuch that for xsatisfyingx(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 …nal state, in …nite time, using the available controls. The reachable set Rof a pointx02M
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 the amount of control e¤ort.
De…nition 2 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)TxM:
The distribution
is called locally …nitely generated if there is a family of vector …elds fXigi=1;m
(called local generators of
) which spans
, i.e.
(x) =spanfX1(x);:::;Xm(x)gTxM. The distrib-
ution
has dimension kifdim
(x) =k, for all points xinM. We recall that the Lie bracket of two
vector …elds is given by
[f;g](x) =@g
@x(x)f(x)@f
@x(x)g(x);
3

(@g
@xis the Jacobian matrix of g). A distribution
onMis 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 …elds 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 …elds, and
therefore it already belongs to
. The Lie brackets are unable to escape
and generate new directions
of motion. We recall that a foliationfSg2AofMis a partition of M=[
2ASofMinto disjoint
connected (immersed) submanifolds S, called leaves.
De…nition 3 A distribution
of constant dimension on Mis called integrable (holonomic) if there
exists a foliation fSg2AonMwhose tangent bundle is
, that isTxS=
(x), whereSis the leaf
passing through x.
Theorem 4 (Frobenius) Suppose that a distribution
has constant dimension. Then,
is integrable if
and only if
is involutive.
De…nition 5 The distribution
=spanfX1;:::;XmgonMis said to be bracket generating if the iterated
Lie brackets
Xi;[Xi;Xj];[Xi;[Xj;Xk]];


;1i;j;km;
span the tangent space TM ofMat every point.
Using the Lie brackets of vector …elds, we construct the ‡ ag of subsheaves


2



r


TM
with

2=
+ [
;
];:::;
r+1=
r+ [
;
r]
where
[
;
r] =spanf[X;Y ] :X2
; Y2
rg:
If there exists an r2such that
r=TM, we say that
is a bracket generating distribution and r
is called the step of the distribution
. In this case the distribution
is not integrable and is called
nonholonomic. This condition is also known as strong Hörmander condition , orLie algebra rank condition .
Ifr= 2the distribution is called strong bracket generating .
The presence of the drift X0in the study of control a¢ ne systems, signi…cantly complicates the
question of controllability. In the following we consider the driftless control a¢ ne system (X0= 0) in the
form
_x=mX
i=1uiXi(x): (3)
4

The vector …elds Xi; i=1;m, generate a distribution
onM(assumed to be connected) such that the
rank of
is constant. Let x0andx1be two points of M. An optimal control problem consists of …nding
those trajectories of the distributional system which connect x0andx1, while minimizing the cost
min
u(
)ZT
0F(u(t))dt; (4)
whereFis a positive homogeneous function on
. We will characterize the controllability using the
properties of vector …elds which generate the distribution
.
Theorem 6 (Chow-Rashevsky) If the distribution
=spanfX1;:::;Xmgis bracket generating (nonholo-
nomic), then the driftless control a¢ ne system is controllable.
If
is not bracket generating and is integrable (holonomic) then the system is not controllable and

determines a foliation on Mwith the property that any curve is contained in a single leaf of the foliation,
and the restriction of
to each leaf of the foliation is bracket generating. We will study in this paper
the case of holonomic distributions.
We assume that the distribution
=spanfX1;X2;:::;Xmgis holonomic with constant rank, which
means that [Xi;Xj]2Dfor everyi;j=1;m,i6=j: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. Next, we will present some notions about Lie algebroids, which are useful in the study
of driftless control a¢ ne systems.
2.2 About Lie Algebroids
LetMbe a real,C1-di¤erentiable, n-dimensional manifold and TxMits tangent space at x2M. The
tangent bundle of Mis denoted (TM;M;M);whereTM=[
x2MTxMandMis the canonical projection
mapM:TM!Mtaking a tangent vector X(x)2TxMTM to the base point x2M:A vector
bundle is a triple (E;;M )whereEandMare manifolds, called the total space and the base space, and
the map:E!Mis a surjective submersion. Using [14] we have:
De…nition 7 A Lie algebroid over a manifold Mis a triple (E;[
;
]E;), where (E;;M )is a vector
bundle of rank moverM;which satis…es the conditions:
a)C1(M)-module of sections (E)is equipped with a Lie algebra structure [
;
]E.
b):E!TM is a bundle map, called the anchor, which induces a Lie algebra homomorphism (also
denoted) from the Lie algebra of sections ((E);[
;
]E)to the Lie algebra of vector …elds ((M);[
;
])
satisfying the Leibniz rule
[s1;fs2]E=f[s1;s2]E+ ((s1)f)s2;8s1;s22(E); f2C1(M):
Also, it results:
1[
;
]Eis aR-bilinear operation,
2[
;
]Eis skew-symmetric, i.e.
[s1;s2]E=[s2;s1]E;8s1;s22(E);
5

3[
;
]Everi…es the Jacobi identity
[s1;[s2;s3]E]E+ [s2;[s3;s1]E]E+ [s3;[s1;s2]E]E= 0;
andbeing a Lie algebra homomorphism, we have
1;s2]E= [(s1);(s2)]:
Iffis a function on M, thendf(x)2E
xis given byhdf(x);ai=(a)f, for8a2Ex. For!2Vk(E)
theexterior derivative dE!2Vk+1(E)is given by the formula
dE!(s1;:::;sk+1) =k+1X
i=1(1)i+1(si)!(s1;:::;^si;:::;sk+1) +
+X
1i<jk+1(1)i+j!([si;sj]E;s1;:::;^si;:::;^sj;:::sk+1):
wheresi2(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 UM, a local basisfsgof the
sections of the bundle 1(U)!Ugenerates local coordinates (xi;y)onE. The local functions i
(x),
L

(x)onMgiven by
(s) =i
@
@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 8 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,i
j=i
j, but if we were to change to another
basis for the vector …elds, the structure functions would become nonzero.
Example 9 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;)(see [13]) with the control space :A!Mwe
mean a section ofEalong. A trajectory of the system is an integral curve of the vector …eld ().
Given the cost function L2C1(A), we have to minimize the integral of Lover the set of those system
trajectories which satisfy certain boundary conditions. The Hamiltonian function H2C1(EMA)is
de…ned by
H(;u) =h;(u)iL (u);
whereas the associated Hamiltonian control system His given by the symplectic equation on Lie algebroid
iH!E=dEH:
6

where!Eis the canonical symplectic form. The critical trajectories are given by [13]
@H
@uA= 0;dxi
dt=i
@H
@;d
dt=i
@H
@xi
L

@H
@: (5)
We can associate to any positive homogeneous cost L:E!Ron Lie algebroids E;a costLon
ImTMde…ned by
L(v) =fL(u)ju2Ex; (u) =vg;
wherev2(Im)xTxM,x2M.
Theorem 10 The relation between the Hamiltonian function Hon the cotangent bundle TMand the
Hamiltonian function Hon the dual Lie algebroid Eis given by
H(p) =H(?(p));  =?(p); p2T
xM; 2E
x: (6)
Proof. The Fenchel-Legendre dual of Lagrangian Lis the Hamiltonian Hgiven by
H(p) = sup
vfhp;viL(v)g= sup
vfhp;viL (u);(u) =vg
= sup
ufhp;(u)iL (u)g= sup
ufh?(p);uiL (u)g=H(?(p));
and we get
H(p) =H();  =?(p); p2T
xM; 2E
x;
or locally
=i
pi; (7)
where the Hamiltonian H(p)is degenerate on Ker?TM: u t
2.3 Applications
Let us consider the following driftless control a¢ ne system with positive homogeneous cost:
8
<
:_x1=u2
_x2=u1+u2x2
_x3=u1+u2x3(8)
min
u(
)ZT
0q
u2
1+u2
2+"u1
dt; 0"<1;
7

We are looking for the optimal trajectories starting from the point (0;1;0)tand parameterized by
arclength (minimum time problem) and free endpoint. The system can be written in the form
_x=u1X1+u2X2; x =0
@x1
x2
x31
A2R3; X1=0
@0
1
11
A; X2=0
@1
x2
x31
A
min
u(
)RT
0F(u(t))dt;F(u) =p
(u1)2+ (u2)2+"u1;0"<1(9)
The vector …elds are given by
X1=@
@x2+@
@x3; X 2=@
@x1+x2@
@x2+x3@
@x3;
The Lie bracket is
[X1;X2] =@
@x2+@
@x3;@
@x1+x2@
@x2+x3@
@x3
=X1:
and it results that the associated distribution
=spanfX1;X2gis holonomic and has the constant rank
2. Moreover, from the system (8) we obtain
_x2_x3= _x1(x2x3);
which yields
lnx2x3=x1+c: (10)
(cis a constant) and it results that
determines a foliation on R3given by the surfaces (10). In order to
solve this optimal control problem we can use the Pontryagin Maximum Principle on the cotangent bundle.
The Lagrangian has the form L=1
2F2(for minimum time problem, every minimizer parametrized by
arclength, or constant speed F= 1is also a minimizer of the so called energy cost L=1
2F2) and we get
the Hamiltonian
H(u;x;p ) =pi_xiL=p1u1+p2(u1+u2x2) +p3(u1+u2x3)1
2p
(u1)2+ (u2)2+"u12
;
The Hamilton-Jacobi-Bellman equations@H
@ui= 0;dxi
dt=@H
@pi;dpi
dt=@H
@xilead to the following system
8
>><
>>:p1+p3x1q
(u1)2+ (u2)2+"u1
"+u1p
(u1)2+(u2)2
= 0
p2q
(u1)2+ (u2)2+"u1
u2p
(u1)2+(u2)2
= 0(11)
and to a very complicated system of implicit di¤erential equations. From (11) is di¢ cult to …nd the
Hamiltonian Hwithout dependence on the control variables. For this reason we will use a di¤erent
approach, involving the framework of Lie algebroids.
8

In order to use the framework of Lie algebroids, we consider E=
(holonomic distribution with
constant rank), the anchor :E!TMis the inclusion and [;]Ethe induced Lie bracket. In the case of
previous example, the anchor has the components
i
=0
@0 1
1×1
1×21
A;
and we consider the Lagrangian function given by
L=1
2p
(u1)2+ (u2)2+"u12
:
Using [9] we can …nd the Hamiltonian on Egiven by
H() =1
2 s
(1)2
(1"2)2+(2)2
1"2"1
1"2!2
: (12)
Using (4) we can calculate the Hamiltonian HonTMin the form H(x;p) =H(); =?(p), where
1
2
=0 1 1
1x1x20
@p1
p2
p31
A:
We get that 1=p2+p3;2=p1+p2x1+p3x2and it results the Hamiltonian on the cotangent bundle
H(x;p) =1
2 s
(p2+p3)2
(1"2)2+(p1+p2x1+p3x2)2
1"2"(p2+p3)
1"2!2
: (13)
Unfortunately, with H(x;p)from (13) the Hamilton’ s equations (1) on TMlead to a complicated system
of di¤erential equations. For this reason, we will use the geometric model of a Lie algebroid. From the
relation [X;X] =L

X
we obtain the non-zero components L1
12= 1; L1
21=1while from (3) we
deduce that
_x1=@H
@2;_x2=@H
@1+x2@H
@2;_x3=@H
@1+x3@H
@2;
_1=1@H
@2;_2=1@H
@1;
where
@H
@1=
1 +"2

1
(1"2)2"q
(1)2
(1"2)2+(2)2
1"2
1"2"2
1
(1"2)3q
(1)2
(1"2)2+(2)2
1"2;
9

@H
@2=2
1"2"12
(1"2)2q
(1)2
(1"2)2+(2)2
1"2:
The form of the last relations leads to the following change of variables
1(t) = (1"2)r(t)sech(t); 2(t) =p
1"2r(t) tanh(t): (14)
where
sinh=ee
2;cosh=e+e
2;tanh=sinh
cosh;sech=1
cosh:
In these conditions we have s
(1)2
(1"2)2+(2)2
1"2=jrj;
and the di¤erential equations
_1=1@H
@2;
with the relations (14) yields
p
1"2_r
r_tanh
=r(tanh+"sechtanh): (15)
Also, from the equation
_2=1@H
@1;
and (14) we get
p
1"2_r
rtanh+_sech2
=r((1 +")2sech2"sech"sech3): (16)
Now, reducing _and_r
rfrom the equations (15) and (16), we obtain
p
1"2_r=r2"sechtanh("sech1);
and p
1"2_=r("sech1)2:
The last two equations lead to
_r
_=r"sechtanh
"sech1;
and respectively to
dr
r="sechtanh
"sech1d;
10

with the solution
lnjrj=ln("sech1)lnc:
Therefore
jrj=1
c("sech1):
Since the optimal trajectories are parameterized by arclength, the conclusion corresponds exactly to the
1=2level of the Hamiltonian and we have
H=r2
2(1"sech)2=1
2c2:
Now,c=1and
r=1
"sech1:
The equation
_1=1_x1;
implies that
x1() = lnc1(1"sech)
(1"2)sech; c 12R:
Since we are looking for the trajectories starting from the point (0;1;0)t, we havex1(0) = 0 and
lnc1
1 +"= 0)c1= 1 +";
which leads to
x1() = ln1"sech
(1")sech= lncosh"
1":
We obtain also that
_2=1
_x2x2@H
@2
=1_x2+x2_1;
and, consequently, 2=1×2+c2. Further,
x2() =sinhp
1"2c2(1"sech)
(1"2)sech:
Fromx2(0) = 1 we obtain that c2= 1 +"and this yields
x2() =sinhp
1"2+cosh"
1":
11

In the same way we get
x3() =sinhp
1"2c3(1"sech)
(1"2)sech:
Fromx3(0) = 0 we obtain that c3= 0and it results
x3() =sinhp
1"2:
Using (8) we have u2= _x1,u1= _x3u2x3= _x2u2x2and by direct computation, we obtain the control
variables
u2() =sinh
cosh"; u 1() =1p
1"21"cosh
cosh":
If"= 0we obtain the case of driftless control a¢ ne systems with quadratic cost with the solution
x1(t) = ln cosht; x2(t) = sinht+ cosht; x3(t) = sinht;
and control variables
u2(t) = tanht; u 1(t) =secht:
Conclusions . In this paper we treat some topics of dynamical systems using Lie geometric methods.
In the case of driftless control a¢ ne systems with holonomic distribution and positive homogeneous cost
we proved that the framework of Lie algebroids is better than cotangent bundles in order to apply the
Pontryagin Maximum Principle and …nd the optimal solution. As futher developments, we try to use
the framework of Lie algebroids in the case of nonholonomic distribution (in particular, strong bracket
generating) and characterize the solutions using the geometry of Lie algebroids.
Acknowledgement.
This research was carried out while the author was visiting the University of North Carolina at
Charlotte (UNCC), USA and was supported by the Horizon 2020 – 2017 RISE – 777911 project. It is a
pleasure to thank to Professor Douglas Shafer (UNCC) for his hospitality and for important discussions
which we had during my stay in Charlotte.
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